Polymath Wiki - User contributions [en]

Hadwiger-Nelson problem

2020-01-21T19:11:37Z

Aubrey: /* Best known results for the chromatic number of spheres */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

The CN in <math>\mathbb{R}^n</math> has been shown to be between <math>(1.239...+o(1))^n</math> and <math>(3+o(1))^n</math>, and a graph construction is known that has <math>8{n \choose 3}</math> vertices and an independence number of <math>\max(6n-28,4n-\max(2,n\mod 4))</math>, which is equivalent to a CN lower bound of <math>2n^2(1+o(1))/9</math>. For specific dimensions, the best known bounds are as follows:

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| 510
| 2508
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://doi.org/10.1007/s00454-014-9612-7 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in 3-space, i.e., 2-dimensional spheres.
Note that the distances given are the 3-dimensional Euclidean distance, not the length of the arc on the sphere surface. Most bounds are due to G.J. Simmons. For a nice summary, see [Malen |https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, the best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for regions of the plane that can be k-colored ==

Four questions of this type have been studied: what proportion of the plane can be tiled with k colours, what is the widest (resp. narrowest) infinite strip that can (resp. cannot) be k-coloured, what is the largest (resp. smallest) circular disc that can (resp. cannot) be k-coloured, and what is the smallest convex shape that cannot be k-coloured (obviously an arbitrarily long but thin rectangle can be 3-coloured).

The results for the proportion of the plane for k=1,2,3,4 are due to Croft; the exact value is k<math>\sqrt{3}\tan(\theta/2)</math> where <math>\theta+\sin(\theta)=\pi/6</math>. The other non-trivial results can be found in the polymath blog. The k=3 lower bound for the strip is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=1 !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| proportion of the plane
| <math>\approx 0.229365\le</math>
| <math>\approx 0.458729\le</math>
| <math>\approx 0.688094\le</math>
| <math>\approx 0.917459\le</math>
| <math>\approx 0.959747\le</math>
| <math>\approx 0.999855\le</math>
|-
| infinite strip, width
| 0
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.4</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| 1
| <math>2/\sqrt{3}\approx 1.155\le,\le 1.25</math>
| <math>\sqrt{(2-1/\sqrt{3})^2 + 1}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>(1+\sqrt{5})\sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485\le</math>
|-
| convex shape, area
| 0
| 0
| <math>\pi/2 \approx 1.5708</math> (semicircle)
|
|
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-21T19:07:16Z

Aubrey: /* Best known results for the chromatic number in higher dimensions */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

The CN in <math>\mathbb{R}^n</math> has been shown to be between <math>(1.239...+o(1))^n</math> and <math>(3+o(1))^n</math>, and a graph construction is known that has <math>8{n \choose 3}</math> vertices and an independence number of <math>\max(6n-28,4n-\max(2,n\mod 4))</math>, which is equivalent to a CN lower bound of <math>2n^2(1+o(1))/9</math>. For specific dimensions, the best known bounds are as follows:

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| 510
| 2508
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://doi.org/10.1007/s00454-014-9612-7 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for regions of the plane that can be k-colored ==

Four questions of this type have been studied: what proportion of the plane can be tiled with k colours, what is the widest (resp. narrowest) infinite strip that can (resp. cannot) be k-coloured, what is the largest (resp. smallest) circular disc that can (resp. cannot) be k-coloured, and what is the smallest convex shape that cannot be k-coloured (obviously an arbitrarily long but thin rectangle can be 3-coloured).

The results for the proportion of the plane for k=1,2,3,4 are due to Croft; the exact value is k<math>\sqrt{3}\tan(\theta/2)</math> where <math>\theta+\sin(\theta)=\pi/6</math>. The other non-trivial results can be found in the polymath blog. The k=3 lower bound for the strip is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=1 !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| proportion of the plane
| <math>\approx 0.229365\le</math>
| <math>\approx 0.458729\le</math>
| <math>\approx 0.688094\le</math>
| <math>\approx 0.917459\le</math>
| <math>\approx 0.959747\le</math>
| <math>\approx 0.999855\le</math>
|-
| infinite strip, width
| 0
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.4</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| 1
| <math>2/\sqrt{3}\approx 1.155\le,\le 1.25</math>
| <math>\sqrt{(2-1/\sqrt{3})^2 + 1}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>(1+\sqrt{5})\sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485\le</math>
|-
| convex shape, area
| 0
| 0
| <math>\pi/2 \approx 1.5708</math> (semicircle)
|
|
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-21T19:04:16Z

Aubrey: /* Best known results for the chromatic number in higher dimensions */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

The CN in <math>\mathbb{R}^n</math> has been shown to be between <math>(1.239...+o(1))^n</math> and <math>(3+o(1))^n</math>, and a graph construction is known that gives a CN of <math>8{n \choose 3}/\max(6n-28,4n-\max(2,n\mod 4))</math>, which is equivalent to <math>2n^2(1+o(1))/9</math>. For specific dimensions, the best known bounds are as follows:

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| 510
| 2508
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://doi.org/10.1007/s00454-014-9612-7 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for regions of the plane that can be k-colored ==

Four questions of this type have been studied: what proportion of the plane can be tiled with k colours, what is the widest (resp. narrowest) infinite strip that can (resp. cannot) be k-coloured, what is the largest (resp. smallest) circular disc that can (resp. cannot) be k-coloured, and what is the smallest convex shape that cannot be k-coloured (obviously an arbitrarily long but thin rectangle can be 3-coloured).

The results for the proportion of the plane for k=1,2,3,4 are due to Croft; the exact value is k<math>\sqrt{3}\tan(\theta/2)</math> where <math>\theta+\sin(\theta)=\pi/6</math>. The other non-trivial results can be found in the polymath blog. The k=3 lower bound for the strip is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=1 !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| proportion of the plane
| <math>\approx 0.229365\le</math>
| <math>\approx 0.458729\le</math>
| <math>\approx 0.688094\le</math>
| <math>\approx 0.917459\le</math>
| <math>\approx 0.959747\le</math>
| <math>\approx 0.999855\le</math>
|-
| infinite strip, width
| 0
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.4</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| 1
| <math>2/\sqrt{3}\approx 1.155\le,\le 1.25</math>
| <math>\sqrt{(2-1/\sqrt{3})^2 + 1}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>(1+\sqrt{5})\sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485\le</math>
|-
| convex shape, area
| 0
| 0
| <math>\pi/2 \approx 1.5708</math> (semicircle)
|
|
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-17T20:00:31Z

Aubrey: /* Best known results for the chromatic number in higher dimensions */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| 510
| 2508
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://doi.org/10.1007/s00454-014-9612-7 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for regions of the plane that can be k-colored ==

Four questions of this type have been studied: what proportion of the plane can be tiled with k colours, what is the widest (resp. narrowest) infinite strip that can (resp. cannot) be k-coloured, what is the largest (resp. smallest) circular disc that can (resp. cannot) be k-coloured, and what is the smallest convex shape that cannot be k-coloured (obviously an arbitrarily long but thin rectangle can be 3-coloured).

The results for the proportion of the plane for k=1,2,3,4 are due to Croft; the exact value is k<math>\sqrt{3}\tan(\theta/2)</math> where <math>\theta+\sin(\theta)=\pi/6</math>. The other non-trivial results can be found in the polymath blog. The k=3 lower bound for the strip is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=1 !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| proportion of the plane
| <math>\approx 0.229365\le</math>
| <math>\approx 0.458729\le</math>
| <math>\approx 0.688094\le</math>
| <math>\approx 0.917459\le</math>
| <math>\approx 0.959747\le</math>
| <math>\approx 0.999855\le</math>
|-
| infinite strip, width
| 0
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.4</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| 1
| <math>2/\sqrt{3}\approx 1.155\le,\le 1.25</math>
| <math>\sqrt{(2-1/\sqrt{3})^2 + 1}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>(1+\sqrt{5})\sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485\le</math>
|-
| convex shape, area
| 0
| 0
| <math>\pi/2 \approx 1.5708</math> (semicircle)
|
|
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-17T19:29:16Z

Aubrey: /* Best bounds for regions of the plane that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://doi.org/10.1007/s00454-014-9612-7 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for regions of the plane that can be k-colored ==

Four questions of this type have been studied: what proportion of the plane can be tiled with k colours, what is the widest (resp. narrowest) infinite strip that can (resp. cannot) be k-coloured, what is the largest (resp. smallest) circular disc that can (resp. cannot) be k-coloured, and what is the smallest convex shape that cannot be k-coloured (obviously an arbitrarily long but thin rectangle can be 3-coloured).

The results for the proportion of the plane for k=1,2,3,4 are due to Croft; the exact value is k<math>\sqrt{3}\tan(\theta/2)</math> where <math>\theta+\sin(\theta)=\pi/6</math>. The other non-trivial results can be found in the polymath blog. The k=3 lower bound for the strip is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=1 !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| proportion of the plane
| <math>\approx 0.229365\le</math>
| <math>\approx 0.458729\le</math>
| <math>\approx 0.688094\le</math>
| <math>\approx 0.917459\le</math>
| <math>\approx 0.959747\le</math>
| <math>\approx 0.999855\le</math>
|-
| infinite strip, width
| 0
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.4</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| 1
| <math>2/\sqrt{3}\approx 1.155\le,\le 1.25</math>
| <math>\sqrt{(2-1/\sqrt{3})^2 + 1}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>(1+\sqrt{5})\sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485\le</math>
|-
| convex shape, area
| 0
| 0
| <math>\pi/2 \approx 1.5708</math> (semicircle)
|
|
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-13T17:14:12Z

Aubrey: /* Best bounds for regions of the plane that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://doi.org/10.1007/s00454-014-9612-7 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for regions of the plane that can be k-colored ==

Three questions of this type have been studied: what is the widest (resp. narrowest) infinite strip that can (resp. cannot) be k-coloured, what is the largest (resp. smallest) circular disc that can (resp. cannot) be k-coloured, and what is the smallest convex shape that cannot be k-coloured (obviously an arbitrarily long but thin rectangle can be 3-coloured).

Most results for the strips are either easy or can be found in the polymath blog. The k=3 lower bound is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| infinite strip, width
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.4</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| <math>2/\sqrt{3}\approx 1.155\le,\le 1.25</math>
| <math>\sqrt{(2-1/\sqrt{3})^2 + 1}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>(1+\sqrt{5})\sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485\le</math>
|-
| convex shape, area
| 0
| <math>\pi/2 \approx 1.5708</math> (semicircle)
|
|
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-13T17:11:08Z

Aubrey: /* Best bounds for regions of the plane that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://doi.org/10.1007/s00454-014-9612-7 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for regions of the plane that can be k-colored ==

Two questions of this type have been studied: what is the widest (resp. narrowest) infinite strip that can (resp. cannot) be k-coloured, and what is the largest (resp. smallest) circular disc that can (resp. cannot) be k-coloured.

Most results for the strips are either easy or can be found in the polymath blog. The k=3 lower bound is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| infinite strip, width
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.4</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| <math>2/\sqrt{3}\approx 1.155\le,\le 1.25</math>
| <math>\sqrt{(2-1/\sqrt{3})^2 + 1}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>(1+\sqrt{5})\sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485\le</math>
|-
| convex shape, area
| 0
| <math>\pi/2 \approx 1.5708</math>
|
|
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-10T23:08:17Z

Aubrey: /* Best bounds for regions of the plane that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for regions of the plane that can be k-colored ==

Two questions of this type have been studied: what is the widest (resp. narrowest) infinite strip that can (resp. cannot) be k-coloured, and what is the largest (resp. smallest) circular disc that can (resp. cannot) be k-coloured.

Most results for the strips are either easy or can be found in the polymath blog. The k=3 lower bound is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| infinite strip, width
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.4</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| <math>2/\sqrt{3}\approx 1.155\le,\le 1.25</math>
| <math>\sqrt{(2-1/\sqrt{3})^2 + 1}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>(1+\sqrt{5})\sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485\le</math>
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-09T04:01:08Z

Aubrey: /* Best bounds for different shapes that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for regions of the plane that can be k-colored ==

Two questions of this type have been studied: what is the widest (resp. narrowest) infinite strip that can (resp. cannot) be k-coloured, and what is the largest (resp. smallest) circular disc that can (resp. cannot) be k-coloured.

Most results for the strips are either easy or can be found in the polymath blog. The k=3 lower bound is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| infinite strip, width
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.25</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| <math>2/\sqrt{3}\approx 1.155\le,\le 1.29</math>
| <math>\sqrt{(2-1/\sqrt{3})^2 + 1}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>(1+\sqrt{5})\sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485\le</math>
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-09T01:50:43Z

Aubrey: /* Best bounds for different shapes that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for different shapes that can be k-colored ==

Most results for the strips are either easy or can be found in the polymath blog. The k=3 lower bound is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| infinite strip, width
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.25</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| <math>2/\sqrt{3}\approx 1.155\le,\le 1.29</math>
| <math>\sqrt{(2-1/\sqrt{3})^2 + 1}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>(1+\sqrt{5})\sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485\le</math>
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-08T21:34:27Z

Aubrey: /* Best bounds for different shapes that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for different shapes that can be k-colored ==

Most results for the strips are either easy or can be found in the polymath blog. The k=3 lower bound is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| infinite strip, width
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.25</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| <math>2/\sqrt{3}\approx 1.155</math>
| <math>\sqrt{(2-1/\sqrt{3})^2 + 1}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>(1+\sqrt{5})\sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485\le</math>
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-08T20:45:46Z

Aubrey: /* Best bounds for different shapes that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for different shapes that can be k-colored ==

Most results for the strips are either easy or can be found in the polymath blog. The k=3 lower bound is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| infinite strip, width
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.25</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| <math>2/\sqrt{3}\approx 1.155</math>
| <math>2\sqrt{(1-1/\sqrt{12})^2 + 1/4}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>(1+\sqrt{5})\sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485\le</math>
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-08T20:45:09Z

Aubrey: /* Best bounds for different shapes that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for different shapes that can be k-colored ==

Most results for the strips are either easy or can be found in the polymath blog. The k=3 lower bound is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| infinite strip, width
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.25</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| <math>2/\sqrt{3}\approx 1.155</math>
| <math>2\sqrt{(1-1/\sqrt{12})^2 + 1/4}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>2\frac{1+\sqrt{5}}{2} \sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485\le</math>
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-08T20:44:38Z

Aubrey: /* Best bounds for different shapes that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for different shapes that can be k-colored ==

Most results for the strips are either easy or can be found in the polymath blog. The k=3 lower bound is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| infinite strip, width
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.25</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| <math>2/\sqrt{3}\approx 1.155</math>
| <math>2\sqrt{(1-1/\sqrt{12})^2 + 1/4}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math>2\frac{1+\sqrt{5}}{2} \sqrt{\frac{15+\sqrt{5}}{10}} \approx 4.2485</math>
|}

Constructions for disks can be found here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-5989
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-08T20:34:38Z

Aubrey: /* Best bounds for different shapes that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for different shapes that can be k-colored ==

Most results for the strips are either easy or can be found in the polymath blog. The k=3 lower bound is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| infinite strip, width
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.25</math>
| <math>9/\sqrt{28}\approx 1.701\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\approx 3.669\le</math>
|-
| disk, diameter
| 1
| <math>2/\sqrt{3}\approx 1.155</math>
| <math>2\sqrt{(1-1/\sqrt{12})^2 + 1/4}\approx 1.739\le</math>
| <math>\approx 2.316\le</math>
| <math></math>
|}

Constructions for disks can be found here:
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
and some older ones here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-08T20:10:24Z

Aubrey: /* Best bounds for different shapes that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for different shapes that can be k-colored ==

Most results for the strips are either easy or can be found in the polymath blog. The k=3 lower bound is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| infinite strip, width
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.25</math>
| <math>9/\sqrt{28}\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\le</math>
|-
| disk, diameter
| 1
| <math>2/\sqrt{3}\approx 1.155</math>
| <math>2\sqrt{(1-1/\sqrt{12})^2 + 1/4}\approx 1.739\le</math>
| <math>2.316\le</math>
| <math></math>
|}

Constructions for disks can be found here:
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
and some older ones here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-08T20:09:53Z

Aubrey: /* Best bounds for different shapes that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for different shapes that can be k-colored ==

Most results for the strips are either easy or can be found in the polymath blog. The k=3 lower bound is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| infinite strip, width
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.25</math>
| <math>9/\sqrt{28}\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\le</math>
|-
| disk, diameter
| 0.5
| <math>2/\sqrt{3}\approx 1.155</math>
| <math>2\sqrt{(1-1/\sqrt{12})^2 + 1/4}\approx 1.739\le</math>
| <math>2.316\le</math>
| <math></math>
|}

Constructions for disks can be found here:
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
and some older ones here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2020-01-08T20:04:46Z

Aubrey: /* Best bounds for different shapes that can be k-colored */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/ Polymath16, thirteenth thread: Bumping the deadline?], Dustin Mixon, July 8, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/ Polymath16, fourteenth thread: Automated graph minimization?], Dustin Mixon, Aug 6, 2019. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/ Polymath16, fifteenth thread: Writing the paper and chasing down loose ends], Dustin Mixon, Dec 12, 2019. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| 553
| 2722
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1907.00929 <math>G_8</math>]
| 529
| 2670
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713 <math>G_8'</math>]
| 529
| 2630
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814 <math>G_9</math>]
| 525
| 2605
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23934<math>G_{10}</math>]
| 517
| 2579
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23999<math>G_{11}</math>]
| '''510'''
| 2508
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf]. For a UD graph on the sphere with many edges, best construction is <math>n\sqrt{\log n}</math> [Swanepoel-Valtr|http://personal.lse.ac.uk/swanepoe/swanepoel-valtr-unitdistance.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Best bounds for different shapes that can be k-colored ==

Most results for the strips are either easy or can be found in the polymath blog. The k=3 lower bound is here https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/#comment-5501. For k=4, see https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24282
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24332
and https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24375.

{| border=1
|-
! Shape !! k=2 !! k=3 !! k=4 !! k=5 !! k=6
|-
| infinite strip, width
| 0
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\approx 0.95876\le,\le 1.25</math>
| <math>9/\sqrt{28}\le</math>
| <math>\sqrt{3}+\sqrt{15}/2\le</math>
|-
| disk, radius
| 0.5
| <math>\sqrt{3}/2\approx 0.866</math>
| <math>\sqrt{(1-1/\sqrt{12})^2 + 1/4}\approx 0.8695\le, \le 1.25</math>
| <math>1.158\le</math>
| <math></math>
|}

Constructions for disks can be found here:
https://dustingmixon.wordpress.com/2019/12/12/polymath16-fifteenth-thread-writing-the-paper-and-chasing-down-loose-ends/#comment-24836
and some older ones here:
https://drive.google.com/file/d/1-LEV4uzd2FjGBvC6cPUL0EDY2cjQy1FB/view
https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4851
upper bound here:
https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24369

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2019-04-18T17:29:15Z

Aubrey: /* Best known results for the chromatic number of spheres */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2010. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| '''553'''
| 2722
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| antipodes
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2019-04-18T16:11:20Z

Aubrey: /* Best known results for the chromatic number of spheres */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2010. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| '''553'''
| 2722
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| unit distances exist
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \frac{\sqrt{23-\sqrt{17}}}{8}=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{23-\sqrt{17}}}{8} < r \le \frac{\sqrt{3-\sqrt 3}}{2}=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\frac{\sqrt{3-\sqrt 3}}{2} < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2019-04-18T05:24:00Z

Aubrey: /* Best known results for the chromatic number of spheres */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2010. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| '''553'''
| 2722
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| unit distances exist
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \sqrt{23-\sqrt{17}}/8=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\sqrt{23-\sqrt{17}}/8 < r \le \sqrt{3-\sqrt 3}/2=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\sqrt{3-\sqrt 3}/2 < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| generalised Moser spindle (needs >2 diamonds for small r)
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser spindle
| unknown
| 8 should be easy, but we want 7
|}

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2019-04-17T21:10:37Z

Aubrey: /* Best known results for the chromatic number of spheres */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2010. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| '''553'''
| 2722
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| unit distances exist
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r \le \sqrt{23-\sqrt{17}}/8=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\sqrt{23-\sqrt{17}}/8 < r \le \sqrt{3-\sqrt 3}/2=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\sqrt{3-\sqrt 3}/2 < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| Moser-spindle
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser-spindle
| unknown
| 8 should be easy, but we want 7
|}

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2019-04-17T20:51:41Z

Aubrey: /* Best known results for the chromatic number of spheres */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2010. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| '''553'''
| 2722
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| unit distances exist
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r < \sqrt{23-\sqrt{17}}/8=0.543..</math>
| 3
| odd cycle
| 4
| tiling given by facets of tetrahedron
|-
| <math>\sqrt{23-\sqrt{17}}/8 \le r < \sqrt{3-\sqrt 3}/2=0.563..</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 4
| tiling given by facets of tetrahedron
|-
| <math>\sqrt{3-\sqrt 3}/2 < r < 1/\sqrt 3</math>
| 3 (4 if measurable)
| odd cycle; probabilistic argument
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| Moser-spindle
| 6
| tiling given by facets of dodecahedron
|-
| <math>\sqrt 3/2 < r</math>
| 4
| Moser-spindle
| unknown
| 8 should be easy, but we want 7
|}

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Probabilistic formulation of Hadwiger-Nelson problem

2019-04-17T01:02:04Z

Aubrey: /* Bounds on p_d for 4-colourings */

Suppose for sake of contradiction that we have a 4-coloring <math>c: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with no unit edges monochromatic, thus

:<math>c(z) \neq c(w) \hbox{ whenever } |z-w| = 1. \quad (1)</math>

We can create further such colorings by composing <math>c</math> on the left with a permutation <math>\sigma \in S_4</math> on the left, and with the (inverse of) a Euclidean isometry <math>T \in E(2)</math> on the right, thus creating a new coloring <math>\sigma \circ c \circ T^{-1}: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with the same property. This is an action of the solvable group <math>S_4 \times E(2)</math>.

It is a fact that all solvable groups (viewed as discrete groups) are [https://en.wikipedia.org/wiki/Amenable_group amenable], so in particular <math>S_4 \times E(2)</math> is amenable. This means that there is a finitely additive probability measure <math>\mu</math> on <math>S_4 \times E(2)</math> (with all subsets of this group measurable), which is left-invariant: <math>\mu(gE) = \mu(E)</math> for all <math>g \in S_4 \times E(2)</math> and <math>E \subset S_4 \times E(2)</math>. This gives <math>S_4 \times E(2)</math> the structure of a finitely additive probability space. We can then define a random coloring <math>{\mathbf c}: {\bf C} \to \{1,2,3,4\}</math> by defining <math>{\mathbf c} := {\mathbf \sigma} \circ c \circ {\mathbf T}^{-1}</math>, where <math>({\mathbf \sigma},{\mathbf T})</math> is the element of the sample space <math>S_4 \times E(2)</math>. Thus for any complex number <math>z</math>, the random color <math>{\mathbf c}(z)</math> is a random variable taking values in <math>\{1,2,3,4\}</math>. The left-invariance of the measure implies that for any <math>(\sigma,T) \in S_4 \times E(2)</math>, the coloring <math> \sigma \circ {\mathbf c} \circ T^{-1}</math> has the same law as <math>{\mathbf c}</math>. This gives the color permutation invariance

:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(z_1) = \sigma(c_1), \dots, {\mathbf c}(z_k) = \sigma(c_k) )\quad (2)</math>

for any <math>z_1,\dots,z_k \in {\bf C}</math>, <math>c_1,\dots,c_k \in \{1,2,3,4\}</math>, and <math>\sigma \in S_4</math>, and the Euclidean isometry invariance
:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(T(z_1)) = c_1, \dots, {\mathbf c}(T(z_k)) = c_k. \quad (3)</math>
(In probabilistic language, this means that the random coloring is a [https://en.wikipedia.org/wiki/Stationary_process stationary process] with respect to the action of <math>S_4 \times E(2)</math>. The extraction of a stationary process from a deterministic object is an example of the ''Furstenberg correspondence principle''.)

== Bounds on <math>p_d</math> for 4-colourings ==

A class of correlations that is of particular interest is that of vertex pairs at some distance <math>d</math>. Accordingly, define
:<math>p_d := {\bf P}( \mathbf{c}(0) = \mathbf{c}(d) ).</math>

{| border=1
|-
! distance !! Lower bound !! Lower-bounding graph/method !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math> \geq 1/2</math>
|
|
| 1/2
| triangle with sides <math>d,d,1</math>
|
|-
| <math>\geq 2/\sqrt{15}</math>
|
|
| <math>15/31</math>
| Proposition 36
|
|-
| <math>\geq (\sqrt{3}-1)/\sqrt{2}</math>
|
|
| <math>12/25</math>
| Proposition 36
|
|-
| large enough
|
|
| <math>323/675 = 0.4785\ldots</math>
| Proposition 36
|
|-
| <math> 0 < d \le \frac{\sqrt{6}-\sqrt{2}}{2} </math>
|<math>1/50</math>
| Lemma 39
|
|
|
|-
| <math> \frac{\sqrt{6}-\sqrt{2}}{2} < d \le \sqrt{\frac{14}{15}}-\frac{1}{\sqrt{15}} </math>
|<math>14/775</math>
| Lemma 39
|
|
|
|-
| <math> \sqrt{\frac{14}{15}}-\frac{1}{\sqrt{15}} < d \le \frac{\sqrt{15}-\sqrt{3}}{4} </math>
|<math>1/100</math>
| Lemma 39
|
|
|
|-
| <math> \frac{\sqrt{15}-\sqrt{3}}{4} < d \le \sqrt{(3-\sqrt{3})\left(1-\frac{1}{\sqrt{2}}\right)} </math>
|<math>1/125</math>
| Lemma 40
|
|
|
|-
| <math>\sqrt{(3-\sqrt{3})\left(1-\frac{1}{\sqrt{2}}\right)} < d \le \sqrt{\frac{19}{15}-\frac{2}{\sqrt{5}}} </math>
|<math>38/3875</math>
| Lemma 40
|
|
|
|-
| <math>\sqrt{\frac{19}{15}-\frac{2}{\sqrt{5}}} < d \le \frac{1}{2}\sqrt{5-2\sqrt{3}} </math>
|<math>1/250</math>
| Lemma 40
|
|
|
|-
| <math> \frac{\sqrt{15}+\sqrt{3}}{4} \le d < \sqrt{\frac{14}{15}}+\frac{1}{\sqrt{15}} </math>
|<math>1/100</math>
| Lemma 39
|
|
|
|-
| <math> \sqrt{\frac{14}{15}}+\frac{1}{\sqrt{15}} \le d < \sqrt{2} </math>
|<math>14/775</math>
| Lemma 39
|
|
|
|-
| <math>\sqrt{2}\le d \le 2</math>
|<math>1/50</math>
| Lemma 39
|
|
|
|-
| <math>2 < d </math>
|<math>1/125</math>
| Lemma 40
|
|
|
|-
| <math> d\geq 1/n, n>1</math>
|
|
| <math>1 - 1/n</math>
| (n+1)-gon with one edge length 1 and the rest d, Lemma 34
|
|-
| <math> d\geq 1/(n \sqrt{3}), n>1</math>
|
|
| <math>(3n-2)/3n</math>
| (n+1)-gon with one edge length <math>1/\sqrt{3}</math> and the rest d, Lemma 34
| Not better than the above on intervals <math>\left(\frac{1}{7},\frac{1}{4\sqrt{3}}\right),\left(\frac{1}{4},\frac{1}{2\sqrt{3}}\right)</math>
|-
| <math>1</math>
| <math>0</math>
| Unit edge
| <math>0</math>
| Unit edge
|
|-
| <math>1/\sqrt{3}</math>
| <math>13/150</math>
| Unit triangle ABC with center O and one vertex at distance <math>1/\sqrt{3}</math> from A,B,O gives <math>6p_{1/\sqrt{3}}+p_{2/\sqrt{3}}\ge 1</math>, upper bound <math>p_{2/\sqrt{3}}\le 12/25</math> finishes proof.
| <math>1/3</math>
| Unit triangle plus its centre
|
|-
| <math>2</math>
| <math>1/4</math>
| <math>G_{21}</math>
| <math>12/25</math>
| Proposition 36
| Lower bound computer verified
|-
| <math>\sqrt{3}</math>
| <math>13/50</math>
| H, Lemma 15+Prop 36
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{7}</math>
|
|
| <math>37/100</math>
| Lemma 38 and Corollary 16
|
|-
| <math>(\sqrt{5}+1)/2</math>
| <math>1/5</math>
| regular pentagon with unit side length
| <math>2/5</math>
| regular pentagon with unit side length
|
|-
| <math>(\sqrt{5}-1)/2</math>
| <math>1/5</math>
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
| <math>2/5</math>
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
|
|-
| <math>\sqrt{11/3}</math>
| <math>1/118</math>
| <math>G_{40}</math> - obsolete
| <math>12/25</math>
| Proposition 36
| computer-verified
|-
| <math>8/3</math>
| <math>1</math>
| <math>V \oplus V \oplus H</math>
| <math>12/25</math>
| Proposition 36
| computer-verified; leads to contradiction
|-
| <math>(\sqrt{6} + \sqrt{2})/2</math>
| <math>13/75</math>
| An arrangement of five vertices: Two unit-length equilateral triangles sharing a vertex, closing angles 90 and 150.
| <math>1/2</math> and | <math>12/25</math>
| Proposition 36
|
|-
| <math>(\sqrt{6} - \sqrt{2})/2</math>
| <math>13/150</math>
| An arrangement of five vertices: Two <math>(\sqrt{6} - \sqrt{2})/2</math>-length equilateral triangles sharing a vertex, closing angles 90 and 150.
| <math>1/2</math> and | <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{2}</math>
| <math>1/14</math>
| A graph of 13 vertices
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math>
| <math>1/28</math>
| A graph of 13 vertices
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{3/2 + \sqrt{33}/6}</math>
| <math>1/196</math>
| A graph of 9 vertices - obsolete
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{5/3}</math>
| <math>1/756</math>
| A graph of 33 vertices - obsolete
| <math>12/25</math>
| Proposition 36
|
|-
| <math>2/\sqrt{3}</math>
| <math>1/177</math>
| A graph of 103 vertices - obsolete
| <math>12/25</math>
| Proposition 36
| Computer-verified
|-
| <math>(\sqrt{33} \pm 1)/(2\sqrt{3})</math>
| <math>1/2</math>
| <math>G_{420}</math>
| <math>12/25</math>
| Proposition 36
| Computer-verified
|}

== Bounds on <math>{\bf P}( \mathbf{c}(0) = \mathbf{c}(d_1) \mid \mathbf{c}(0) \neq \mathbf{c}(d_0) )</math> for 4-colourings ==

{| border=1
|-
! <math>d_1</math> !! <math>d_0</math> !! Lower bound !! Lower-bounding graph !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math>1+(-1)^{1/3}</math>
| <math>2</math>
| 1/2
| H
|
|
| Equals <math>p_{\sqrt 3}/(1-p_2)</math>
|-
| <math>1+(-1)^{1/3}</math>
| <math>1+(-1)^{-1/3}</math>
|
|
| 1/2
| H
|
|}

== Proofs of bounds ==

One can compute some correlations of the coloring exactly:

=== Lemma 1 ===
Let <math>z,w \in {\bf C}</math> with <math>|z-w|=1</math>. Then
:<math> {\bf P}( \mathbf{c}(z) = c ) = \frac{1}{4}\quad (4)</math>
for all <math>c=1,\dots,4</math>,
:<math> {\bf P}( \mathbf{c}(z) = \mathbf{c}(w) ) = 0\quad (5),</math>
and
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c' ) = \frac{1}{12} \quad (6)</math>
for any distinct <math>c,c' \in \{1,2,3,4\}</math>. If <math>u</math> is at a unit distance from both z and w, then
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c'; \mathbf{c}(u) = c'' ) = \frac{1}{24} \quad (6')</math>

'''Proof''' By color invariance (2), the four probabilities in (4) are equal and sum to 1, giving (4). The claim (5) is immediate from (1). From (5) and color invariance, the 12 probabilities in (6) are equal and sum to 1, giving (6). The same argument gives (6').<math>\Box</math>

=== Lemma 2 ===
(Spindle argument) Let <math>|d| \geq 1/2</math>. Then
:<math> p_d \leq \frac{1}{2} \quad (7).</math>

'''Proof''' We can find an angle <math>\theta</math> with <math>|de^{i\theta}-d|=1</math>, then <math>\mathbf{c}(de^{i\theta}) \neq \mathbf{c}(d)</math> almost surely. This means that at least one of the events <math>\mathbf{c}(0) = \mathbf{c}(d)</math>, <math>\mathbf{c}(0) = \mathbf{c}(d e^{i\theta})</math> occurs with probability at most 1/2. The claim now follows from isometry invariance (3). <math>\Box</math>

=== Lemma 3 ===
(Using the K graph) We have
:<math>52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) + {\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} ) \geq 1 \quad (8).</math>

'''Proof''' Consider the 61-vertex graph <math>K</math> from [https://arxiv.org/abs/1804.02385 de Grey's paper]. It has 26 (isometric) copies of H, and thus 52 copies of the triangle <math>(1, e^{2\pi i/3}, e^{4\pi i/3})</math>. With probability at least <math>1 - 52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) </math>, none of these triangles are monochromatic. By the argument in that paper, this implies that the three linking diagonals <math>(-2, +2)</math>, <math>(-2 e^{2\pi i/3}, 2e^{2\pi i/3})</math>, <math>(-2 e^{4\pi i/3}, e^{-4\pi i/3})</math> are monochromatic. This gives the claim. <math>\Box</math>

=== Corollary 4 ===
(Existence of monochromatic <math>\sqrt{3}</math>-triangles) We have <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) \geq \frac{1}{104}</math>.

'''Proof''' The probability <math>{\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} )</math> is at most <math>{\bf P}( \mathbf{c}(-2) = \mathbf{c}(2)) = p_4</math>, which by Lemma 2 is at most 1/2. The claim now follows from Lemma 3. <math>\Box</math>

=== Computer-verified Claim 5 ===
(Using the graph M) One has <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = 0</math> (Note this contradicts Corollary 4).

'''Proof''' This simply reflects the fact that there is no 4-coloring of the 1345-vertex graph M from [https://arxiv.org/abs/1804.02385 de Grey's paper] with its central copy of H containing a monochromatic triangle. One can use other graphs for this purpose, such as the 278-vertex graph <math>M_1</math> or <math>V \oplus V \oplus H</math>. <math>\Box</math>

=== Computer-verified Claim 6 ===
(Using the graph <math>V \oplus V \oplus H</math>) One has <math>p_{8/3} = 1</math> (note this contradicts Lemma 2).

'''Proof''' This reflects the fact that every 4-coloring of <math>V \oplus V \oplus H</math> must assign the same color to 0 and 8/3. There is also a 745-vertex subgraph of <math>V \oplus V \oplus H</math> with the same property. <math>\Box</math>

=== Lemma 7 ===
(Using <math>G_{40}</math>) We have
:<math>59 p_{\sqrt{11/3}} + p_{8/3} \geq 1</math>.

'''Proof''' This reflects the fact that every 4-coloring of the 40-vertex graph <math>G_{40}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which none of the 59 pairs of vertices at distance <math>\sqrt{11/3}</math> apart, will assign the same color to 0 and 8/3. (This is presumably human-verifiable.) <math>\Box</math>

=== Corollary 8 ===
One has
:<math> p_{\sqrt{11/3}} \geq \frac{1}{118}</math>.

'''Proof''' Combine Lemma 7 and Lemma 2. <math>\Box</math>

=== Lemma 9 ===
(Using <math>G_{49}</math>) One has
:<math>18 {\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq p_{\sqrt{11/3}} .</math>

'''Proof''' This reflects the fact that every 4-coloring of the 49-vertex graph <math>G_{49}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which 0 and <math>\sqrt{11/3}</math> have the same color, at least one of the 18 copies of <math>(1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3)</math> is monochromatic. This is potentially human-verifiable. <math>\Box</math>

=== Corollary 10 ===
(Existence of monochromatic <math>1/\sqrt{3}</math>-triangles) One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq \frac{1}{2124}.</math>

'''Proof''' Combine Corollary 8 and Lemma 9. <math>\Box</math>

=== Computer-verified claim 11 ===
One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) = 0</math>. (This contradicts Corollary 10).

'''Proof''' This reflects the fact that the 627-vertex graph <math>G_{627}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] does not have any 4-colorings with <math>1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3</math> monochromatic. <math>\Box</math>

=== Lemma 12 ===
For certain special distances d, one can improve the bound in Lemma 2:

If <math>k \geq 1</math> is a natural number, <math>j\in\mathbb{Z}</math>, <math>\gcd(j,2k+1)=1</math>, and <math>r = \frac{1}{2} \csc\left(\frac{j\pi}{2k+1}\right)</math> then
:<math> p_r \leq \frac{k}{2k+1},</math>
thus for instance
:<math> p_{\frac{1}{\sqrt{3}}} \leq \frac{1}{3}.</math>

'''Proof''' Observe that the regular 2k+1-polygon <math>r, re^{2\pi i/(2k+1)}, r e^{4\pi i/(2k+1)}, \dots, r^{4k\pi i/(k+1)}</math> has unit side lengths. By the pigeonhole principle, we conclude that at most k of these vertices can have the same color as the origin, and the claim follows. <math>\Box</math>

On the other hand, for <math>k=2,j=1</math> we also know from the regular pentagon of unit sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}+1}{2}} \leq \frac{2}{5} \quad (9)</math>
since any 4-coloring of that pentagon has either one or two monochromatic diagonals.

Similarly, for <math>k=2,j=2</math> we also know from the regular pentagon of <math>\frac{\sqrt{5}-1}{2}</math> sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}-1}{2}} \leq \frac{2}{5}</math>
since any 4-coloring of that pentagon has either one or two monochromatic edges. More generally, if <math>a,b,c,d,e</math> are the diagonal lengths of a pentagon with unit sides, then
:<math> 1 \leq p_a + p_b + p_c + p_d + p_e \leq 2.</math>

=== Lemma 13 ===
We have
:<math> 7 p_{\frac{1}{\sqrt{3}}} \geq p_{\sqrt{3}}</math>.

'''Proof''' Consider the unit rhombus <math>0, 1, e^{i\pi/3}, e^{-i\pi/3}</math> together with the centers <math>e^{i\pi/6}/\sqrt{3}, e^{-i\pi/6}/\sqrt{3}</math>. With probability <math>p_{\sqrt{3}}</math>, the two far vertices <math>e^{i\pi/3}, e^{-i\pi/3}</math> are the same color, and then 0,1 will be two other colors. This forces either one of the centers <math>e^{i\pi/6}/\sqrt{3}</math> of a triangle to have a common color with one of the vertices of that triangle, or the two centers must have the same color. Thus in any event one of the seven edges of distance <math>1/\sqrt{3}</math> is monochromatic, giving the claim. <math>\Box</math>

=== Corollary 14 ===
We have <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{728}</math>.

This slightly improves upon the lower bound of 1/2124 coming from Corollary 10.

'''Proof''' Combine Corollary 4 and Lemma 13. <math>\Box</math>

=== Lemma 15 ===
One has
:<math>p_{\sqrt{3}} + p_2 \geq \frac{2}{3}</math> and <math>2 p_{\sqrt{3}} + p_2 \geq 1</math>.

'''Proof''' As noted in de Grey's paper, there are essentially four 4-colorings of H. H has six edges of length <math>\sqrt{3}</math> and three of length <math>2</math>. If we let a denote the number of monochromatic <math>\sqrt{3}</math> edges and b the number of monochromatic <math>2</math>-edges, we see from inspection of all four colorings that <math>(a,b)</math> is either <math>(6, 0), (4,0), (2, 1)</math>, or <math>(0,3)</math>. In particular, one always has <math>\frac{a}{6} + \frac{b}{3} \geq \frac{2}{3}</math> and <math>2\frac{a}{6} + \frac{b}{3} \geq 1</math>. Taking expectations, we obtain the claim. <math>\Box</math>

=== Corollary 16 ===
We have <math>p_2 \geq \frac{1}{6}</math>, <math>p_{\sqrt{3}} \geq \frac{1}{4} </math> and <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{28}</math>.

'''Proof''' Combine Lemma 2, Lemma 15, and Lemma 13. <math>\Box</math>

=== Lemma 17 ===
If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths a,b,c. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also, thus
:<math>{\bf P}(\mathbf{c}(0) \neq \mathbf{c}(a)) + {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(b)) \geq {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(c))</math>
and the claim follows. <math>\Box</math>

Note that Lemma 2 follows from the a=b, c=1 case of this lemma. Iterating this lemma starting with Lemma 2 we can also obtain slightly nontrivial upper bounds on <math>p_a</math> for small values of a, e.g. <math>p_a \leq 1 - 2^{-k}</math> when <math>a \geq 2^{-k}, k\in\mathbb{Z}^+</math>.

Further, we can generalise the a=b case to one in which the triangle is replaced by a (k+1)-gon of which one edge is 1 and the others are all equal, leading to the stronger result <math>p_a \leq 1 - 1/k</math> when <math>a \geq 1/k, k\in\mathbb{Z}^+ \land k>1</math>. Further strengthening is achieved by using <math>1/\sqrt{3}</math> as the long edge, given Lemma 12.

=== Lemma 18 ===
Whenever <math>d>0</math>, one has the inequalities
:<math> |p_{\phi d} - p_d| \leq \frac{2}{5}, p_{\phi d} + p_d \geq \frac{1}{5}, 2p_d - p_{\phi d} \leq 1, 2 p_{\phi d} - p_d \leq 1</math>
where <math>\phi := \frac{1+\sqrt{5}}{2}</math> is the golden ratio. Also we have
:<math> p_{d/\sqrt{3}} \leq \frac{1}{3} + p_d, \frac{1}{2} + \frac{1}{2} p_d.</math>

Note that this generalises (9), as well as a special case of Lemma 12.

'''Proof''' Consider the regular pentagon with sidelength <math>d</math>, so it also has 5 diagonals of length <math>\phi d</math>. Let <math>a \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic edges and let <math>b \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic diagonals. Observe:
* <math>a,b</math> cannot both be zero (pigeonhole principle).
* <math>a</math> cannot be 4. Similarly, <math>b</math> cannot be 4.
* If <math>a=5</math>, then <math>b=5</math>, and conversely.
* If <math>a=0</math>, then <math>b=1,2</math>; similarly, if <math>b=0</math>, then <math>a=1,2</math>.

From this we observe the inequalities
:<math> |\frac{a}{5}-\frac{b}{5}| \leq \frac{2}{5}; \frac{a}{5} + \frac{b}{5} \geq \frac{1}{5}; 2 \frac{a}{5} - \frac{b}{5} \leq 1; 2\frac{b}{5} - \frac{a}{5} \leq 1</math>
and on taking expectations we obtain the first claim. Similarly, if one considers the colorings of an equilateral triangle of sidelength <math>d</math> together with its center, and counts the numbers <math>a,b \in \{0,1,2,3\}</math> of monochromatic edges of length <math>d</math> and <math>d/\sqrt{3}</math> respectively, one observes that one always has <math>\frac{b}{3} \leq \frac{1}{3} + \frac{2}{3} \frac{a}{3}, \frac{1}{2} + \frac{1}{2} \frac{a}{3}</math>, and on taking expectations one obtains the claim.<math>\Box</math>

The hexagon <math>H</math> has essentially four distinct colorings: the coloring <math>\hbox{2tri}</math> with two triangles, the coloring <math>\hbox{1tri}</math> with one triangle, the coloring <math>\hbox{axisym}</math> that is symmetric around an axis, and the coloring <math>\hbox{centralsym}</math> that is symmetric around the central point. This gives four probabilities <math>p_{H = 2tri}, p_{H = 1tri}, p_{H = axisym}, p_{H = centralsym}</math> that sum to 1. By counting the number of monochromatic edges of length <math>\sqrt{3}, 2</math> respectively, one also obtains the identities
:<math>p_{\sqrt{3}} = p_{H = 2tri} + \frac{2}{3} p_{H = 1tri} + \frac{1}{3} p_{H = axisym}; \quad p_2 = \frac{1}{3} p_{H=axisym} + p_{H=centralsym}</math>
which reproves Lemma 15. Also
:<math> {\bf P}( \mathbf{c}(0) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = p_{H = 2tri} + \frac{1}{2} p_{H=1tri}.</math>
Any 4-coloring of L contains at least one triangle within one of its 52 copies of H, thus
:<math> p_{H = 2tri} + \frac{1}{2} p_{H=1tri} \geq \frac{1}{52}</math>
which reproves Corollary 4.

=== Lemma 19 ===
(Hubai) One has <math>p_{H = 1tri} + p_{H = axisym} \geq \frac{1}{10}</math>.

'''Proof''' Consider five copies of H centred at 0,1,2,3,4. With probability at least <math>1 - 5( p_{H = 1tri} + p_{H = axisym} )</math>, none of these copies of H are colored 1tri or axisym, and so must be colored 2tri or centralsym. One can check then that if one of the copies is colored 2tri, then so is any adjacent copy; thus all five copies are colored 2tri, or all five are colored centralsym. In either case we see that -1 and 5 are colored the same color. Comparing with Lemma 2 then gives the claim. <math>\Box</math>

=== Theorem 20 ===
One has <math>p_{H = 1tri} > 0</math>.

'''Proof''' Suppose for contradiction that <math>p_{H = 1tri} = 0</math>. One can then run a version of the de Bruijn-Erdos argument to obtain a coloring in which 1tri hexagons are completely nonexistent (since there are arbitrarily large finite colorings with this property). Consider the triangular lattice <math>{\bf Z}[e^{\pi i/3}]</math>. We 2-color the edges of this lattice by coloring an edge black if it is the short diagonal of a unit rhombus with monochromatic long diagonal, and white otherwise. The four colorings of hexagons lead to four possible colorings at each vertex:

* If H is colored 2tri, then all six edges to the centre of H are black.
* If H is colored 1tri, then two edges to the centre of H at 120 degree angles are white, the other four are black.
* If H is colored axisym, then two opposing edges of the centre of H are black, the other four are white.
* If H is colored centralsym, then all six edges to the centre of H are black.

In particular, as we are assuming no 1tri hexagons, the faces cut out by the black edges have angles 60 degrees, and thus must be equilateral triangles, sectors of angle 60, half-planes, or the entire plane. If there is at least one equilateral triangle, then the rest of the black edges must form an equilateral lattice with that triangle sidelength. This leads to only a small number of possible hexagon colorings in the lattice:

# Case 1: All edges white.
# Case 2: All edges black.
# Case 3.k: For some natural number <math>k \geq 2</math>, the length k edges joining adjacent vertices in some coset of <math>k \cdot {\mathbf Z}[ e^{\pi i/3} ]</math> are all black, and the remaining edges are white.
# Case 4: Each horizontal row consists either entirely of black edges, or entirely of white edges.
# Case 5: Each northwest row consists either entirely of black edges, or entirely of white edges.
# Case 6: Each northeast row consists either entirely of black edges, or entirely of white edges.
# Case 7: Six rays of black edges meeting at a common vertex; all other edges white.

Technically, Case 1 is contained in Cases 4,5,6 as written above, but this will not be an issue. One can view Case 7 as a limiting case <math>k=\infty</math> of Case 3.k; Case 2 is similarly the opposite limiting case <math>k=1</math>.

In the first case, the coloring is periodic with periods <math>2, 2 e^{\pi i/3}</math>. In the second case, it is periodic with periods <math>3, 3 e^{\pi i/3}</math>. In the third case, it is periodic with periods <math>3k, 3k e^{\pi i/3}</math>. Also note that for each k, one can check if Case 3.k holds by inspecting the coloring at a finite number of vertices. Thus the event that Case 3.k holds is "measurable" in the sense that a meaningful probability can be assigned. (But Cases 1,2,4,5,6 are not measurable events, they require an infinite number of points to be inspected, and the probability measure we are using is only finitely additive rather than infinitely additive.) In Case 4, the coloring is periodic with period 2; also, every coset of <math>2 \cdot {\bf Z}[e^{\pi i/3}]</math> is 2-colored. Similarly for Case 5 and 6 (where the periods are <math>2 e^{2\pi i/3}</math> and <math>2 e^{4\pi i/3}</math> respectively.)

Let <math>\alpha_k</math> be the probability that Case 3.k holds for the given value of <math>k</math>. Then <math> \sum_{k=2}^K \alpha_k \leq 1</math> for any k, hence <math>\sum_{k=2}^\infty \alpha_k \leq 1</math>. In particular, we can find <math>K_1</math> such that <math>\sum_{k={K_1}}^\infty \alpha_k \leq 0.1</math> (say). Let <math>P</math> be six times the least common multiple of <math>1,2,\dots,K_1</math>. Then the coloring is P- and <math>P e^{\pi i/3}</math>-periodic for Case 1, Case 2, and all Case 3.k with <math>k \leq K_1</math>. On the other hand, if <math>K_2</math> is sufficiently large depending on <math>P</math>, and Case 3.k holds for some <math>k \geq K_2</math>, then almost all of the hexagons are colored centralsym, which makes the coloring "almost <math>P, P e^{\pi i/3}</math>-periodic" in the sense that
:<math>{\bf c}(z+P e^{\pi i j/3}) = {\bf c}(z) \hbox{ for } j=0,1,2,3,4,5</math>
will hold for at least <math>0.9</math> of the lattice points <math>z \in {\bf Z}[e^{\pi i/3}]</math> with <math>|z| \leq K_2</math>. Similarly for Case 7 (which is sort of a <math>k=\infty</math> limiting case of Case 3.k.) Thus, with the probability <math> \geq 1 - \sum_{k=K_1}^{K_2} \alpha_k \geq 0.9</math>, the coloring of the seven vertices <math>{\bf c}(0), {\bf c}(P e^{\pi ij/3}, j=1,\dots,6</math> is (up to rotation and recoloring) one of the three patterns of the central and linking vertices in Figure 3 of Aubrey's paper, namely

* <math>{\bf c}(0) = {\bf c}(P) = {\bf c}(P e^{\pi i/3}) = {\bf c}(P e^{2\pi i/3}) = {\bf c}(P e^{3\pi i/3}) = {\bf c}(P e^{4\pi i/3}) = {\bf c}(P e^{5\pi i/3}) </math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{3\pi i/3})</math>; <math>{\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{3\pi i/3})</math>; <math> {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>.

Using the spindling argument from Aubrey's paper, we conclude that the third possibility must in fact hold with probability at least 0.8; on the other hand, from Lemma 2 this scenario can only occur with probability at most 1/2, giving the required contradiction.
<math>\Box</math>

One should be able to refine this argument to show that <math>p_{H = 1tri} > c</math> for an absolute constant <math> c>0 </math>.

=== Lemma 21 ===
Providing a tighter bound for Lemma 17 with a more thorough proof: If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths <math>a,b,c</math>. Let <math>\left|z_2\right|=b,\left|a-z_2\right|=c</math>. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also: <math>\mathbf{c}(a)\neq\mathbf{c}(z_2)\Rightarrow[\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)]</math>. <math>[A\Rightarrow B]\Rightarrow {\bf P}(A)\leq{\bf P}(B)</math> thus
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) \geq {\bf P}(\mathbf{c}(a) \neq \mathbf{c}(z_2)) = 1-p_c</math>
<math>{\bf P}(A\lor B) +{\bf P}(A\land B)={\bf P}(A)+{\bf P}(B)</math>, so
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)) + {\bf P}(\mathbf{c}(0)\neq\mathbf{c}(z_2)) - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>1-p_c \leq 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
Using the law of cosines: <math>z_2=b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)</math>
:<math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math> <math>\Box</math>

=== Lemma 22 ===

One has <math>3 p_{1/\sqrt{3}} \geq {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) )</math>.

'''Proof''' Let <math>z</math> be a complex number of magnitude <math>1/\sqrt{3}</math> that is a unit distance from 1. If <math>\mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) = c</math> (say), then <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> cannot be colored with <math>c</math>; also, <math>z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> are the vertices of a unit equilateral triangle and thus must take on three different colors. By the pigeonhole principle, one of <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> must then take the same color as the origin, and the claim follows. <math>\Box</math>

=== Lemma 23 ===

One has <math>4 p_{(\sqrt{6} \pm \sqrt{2})/2} + p_{\sqrt{2}} \geq 1</math>. In particular, by Lemma 2, <math>p_{(\sqrt{6}+\sqrt{2})/2} \geq 1/8</math>.

'''Proof''' [ExIs2018b] We just prove the claim for the + sign (the - sign can then be obtained after applying the Galois conjugacy that maps <math>\sqrt{3}</math> to <math>-\sqrt{3}</math>, leaving <math>\sqrt{2}</math> unchanged). Set <math>d := \frac{\sqrt{6}+\sqrt{2}}{2}</math>, and consider the five vertices

:<math>0, e^{5\pi i/4}, e^{5\pi i/4} + d, e^{5\pi i/4} + e^{\pi i/3} d, e^{5\pi i/4} + (e^{\pi i/3}-i)d.</math>

One can check that of the ten edges determined by these five vertices, five have unit length, four have length d, and the remaining distance (from 0 to <math>e^{5\pi i/4}+d</math>) has distance <math>\sqrt{2}</math>. Since <math>\mathbf{c}</math> must make one of the latter five edges monochromatic, the claim follows. <math>\Box</math>

=== Lemma 24 ===

One has <math>p_{\sqrt{2}} \geq \frac{1}{14}</math>.

'''Proof''' In page 7 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 20 unit distance edges and 14 edges of length <math>\sqrt{2}</math> is constructed. The coloring <math>\mathbf{c}</math> must make one of the 14 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 25 ===

Let <math>d = \frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math> and <math>e = \frac{3^{1/4} \sqrt{2} + \sqrt{3} - 1}{2}</math>.
Then one has <math>14 p_d + p_e \geq 1</math>. In particular, by Lemma 2, <math>p_d \geq 1/28</math>.

'''Proof''' In page 9 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 19 unit edges, 14 edges of length d, and one edge of length e is constructed. The coloring <math>\mathbf{c}</math> must make one of the 15 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 26 ===

Let <math>d = \sqrt{3/2 + \sqrt{33}/6}</math>. Then <math>7 p_d \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_d \geq \frac{1}{196}</math>.

'''Proof''' In page 11 of [ExIs2018b], a graph of nine vertices consisting of 12 unit edges and 7 edges of length d is constructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Thus, <math>\mathbf{c}</math> can only make the AB edge monochromatic if one of the seven length d edges is monochromatic. The claim follows.
<math>\Box</math>

=== Lemma 27 ===

One has <math>27 p_{\sqrt{5/3}} \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_{\sqrt{5/3}} \geq \frac{1}{756}</math>.

'''Proof''' In page 13 of [ExIs2018], a graph of 33 vertices with some unit edges and 27 edges of length <math>\sqrt{5/3}</math> is contructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Now repeat the proof of Lemma 26. <math>\Box</math>

=== Computer-verified claim 28 ===

One has <math>p_{2/\sqrt{3}} \geq \frac{1}{177}</math>.

'''Proof''' In page 15 of [ExIs2018], a 5-chromatic graph of 103 vertices, 312 unit edges, and 177 edges of length <math>2/\sqrt{3}</math> is constructed. <math>\mathbf{c}</math> must make one of the latter edges monochromatic, giving the claim. <math>\Box</math>

=== Lemma 29 ===

One has <math>p_{(\sqrt{6} \pm \sqrt{2})/2} \geq 1/6</math> (this improves the bound in Lemma 23).

'''Proof''' Use graphs 505 and 507 from [S2004] and the spindle bound. <math>\Box</math>

=== Lemma 30 ===
For <math>m > n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' <math>n</math> colors and <math>m</math> points necessitates at least 2 having equal color. I.e.

:<math>{\bf P}\left(\bigvee_{k=0}^n \bigvee_{j=k+1}^n\ \mathbf{c}(z_k) = \mathbf{c}(z_j)\right) = 1</math>

The lemma then follows immediately from the fact:
:<math>{\bf P}\left(\bigcup_{k} E_k\right) \leq \sum_{k} {\bf P}\left(E_k\right) \,\Box</math>

=== Corollary 31 ===
Let <math>\lvert z_k\rvert=1</math>. Then for <math>m \geq n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' Use lemma 30 on the set <math>\left\{z_k \bigg\vert 1\leq k\leq m \land k\in\mathbb{Z}\right\}\cup\{0\}</math>. Simplify using <math>{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(0) \right)=0</math>.
<math>\Box</math>

=== Lemma 32 ===
For <math>x\in\mathbb{R}</math> and an <math>n</math>-coloring of the plane, <math>\sum_{k=1}^{n-1}\left(n-k\right){\bf P}\left(\mathbf{c}\left(0\right) = \mathbf{c}\left( 2\sin\left(\frac{kx}{2}\right) \right) \right) \geq 1</math>.

'''Proof''' Use corollary 31 on the set <math>\left\{e^{ikx} \bigg\vert 0\leq k < n \land k\in\mathbb{Z}\right\}</math>. and simplify by grouping lengths.
<math>\Box</math>

=== Corollary 33 ===
Interesting(easy to simplify results of) values for <math>x</math> in Lemma 32 are in <math>\left\{x \bigg\vert \sin\left(\frac{kx}{2}\right)=1 \land 1\leq k < n \land k\in\mathbb{Z}\right\}</math>.
For 4-colorings, this gives
:<math>2p_{\sqrt 3}+p_2 \geq 1</math>
:<math>3p_{(\sqrt 3-1)/\sqrt 2}+p_{\sqrt 2} \geq 1</math>
:<math>3p_{2\sin(\pi/18)}+2p_{2\sin(\pi/9)} \geq 1</math>

=== Lemma 34 ===
Generalizing the note of Lemma 17, <math>\lvert d_1\rvert= d_1 > \lvert d_0\rvert= d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>

'''Proof''' let <math>\lvert z_{j+1} -z_j\rvert=d_0 > 0, \lvert z_{j+n} -z_0\rvert=d_1>0</math>.

Base case, <math>n=2</math>, by Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle:
:<math>2d_0\geq d_1\Rightarrow 2p_{d_0}\leq 1+p_{d_1}</math>
The inductive step is Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle. After induction:
:<math>[n\geq 2\land nd_0\geq d_1]\Rightarrow np_{d_0}\leq n-1+p_{d_1}</math>
Substitute <math>n=\left\lceil\frac{d_1}{d_0}\right\rceil</math>, simplify, rearrange:
:<math>d_1 > d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>
<math>\Box</math>

=== Proposition 35 ===

If <math>d > 1/\sqrt{2}</math> obeys the inequalities
:<math>\sin( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
and
:<math>\cos( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
then
:<math>p_d \leq \frac{1}{2} - \frac{1}{188}.</math>

(One can check that the conditions are obeyed precisely when <math>d \geq \frac{\sqrt{33}-1}{8} = 0.84307\dots</math>.)

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. Since <math>AB</math> is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the triangle <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>.

Now let <math>BADE</math> be a rhombus with sidelengths d and <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. By the hypotheses, the diagonals BD, AE of this rhombus have length at least 1/2, and hence are monochromatic with probability at most 1/2 by Lemma 2. As above, ABD and BDE are each monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>. As BD is monochromatic with probability at most <math>1/2</math>, we conclude that BADE is monochormatic with probability at least <math>\frac{1}{2}-6\delta</math>.

Now let <math>EDFG</math> be another rhombus congruent to <math>BADE</math>. As BD, AE have length at least 1/2, at least one of the long diagonals BF, AG have length at least 1/2 (the diagonal opposite an obtuse or right-angled triangle will work). Let's say BF has length at least 1/2. As BADE and EDFG are both monochromatic with probability at least <math>\frac{1}{2}-6\delta</math>, and the common edge DE is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the entire configuration ABDEFG is monochromatic with probability at least <math>\frac{1}{2}-11\delta</math>. In particular the pentagon ABDEF is monochromatic with at least this probability. However, in this pentagon, the five edges BA, AD, DE, EB, EF are monochromatic with probability at most <math>\frac{1}{2}-\delta</math>, and the other five edges are monochromatic with probability at most <math>\frac{1}{2}</math> by Lemma 2. Thus the probability that at least one of the edges of this pentagon is monochromatic is at most <math>(\frac{1}{2}-11\delta) + 5 \times 10\delta + 5 \times 11\delta = \frac{1}{2}+94\delta</math>. On the other hand, by the pigeonhole principle, this probability is 1. The claim follows. <math>\Box</math>

=== Proposition 36 ===

If <math>d \ge \frac{2}{\sqrt{15}} = 0.5163\dots</math>, then <math>p_d \leq \frac{1}{2} - \frac{1}{62}</math>,

if <math>d \ge \frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math>, then <math>p_d \leq 0.48</math>,

and <math>\limsup_d p_d\leq \frac{323}{675}=0.4785\ldots</math> (so <math>p_d<0.4786</math> if <math>d</math> is large enough).

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. A simple calculation shows that if <math>d \ge \frac{2}{\sqrt{15}}</math>, then <math>|BD| \ge \frac{1}{2}</math>.
If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. By inclusion-exclusion, we conclude that outside of the event that <math>AB</math> is monochromatic, the probability that <math>AD</math> is monochromatic is at most <math>2\delta</math>.

Now let <math>ADE</math> the <math>180^\circ</math> rotation of <math>ADB</math> around the midpoint of <math>AD</math>, and let <math>FDE</math> be the <math>180^\circ</math> rotation of <math>ADE</math> around the midpoint of <math>DE</math>. By the hypotheses, the line segments <math>AE, BD, BE, BF, DF</math> all have length at least 1/2. Let <math>{\mathcal E}</math> be the event that at least one of <math>AB, AD, DE, EF</math> is monochromatic. By the previous paragraph, this event occurs with probability at most <math>\frac{1}{2}-\delta+2\delta+2\delta+2\delta = \frac{1}{2}+5\delta</math>.

By previous considerations, <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>, and this event lies in <math>{\mathcal E}</math>. On the other hand, <math>BD</math> is monochromatic with probability at most 1/2 by Lemma 2. We conclude that outside of <math>{\mathcal E}</math>, <math>BD</math> is only monochromatic with probability at most <math>3\delta</math>. A similar argument (replacing <math>ABD</math> by <math>DAE</math> or <math>EDF</math>) shows that outside of <math>{\mathcal E}</math>, <math>AE</math> is monochromatic with probability at most <math>3\delta</math>, and similarly for <math>DF</math>. Now we consider <math>EB</math>. By previous considerations, the probability that <math>ABDE</math> is monochromatic is at least <math>\frac{1}{2}-5\delta</math>, and this event lies inside <math>{\mathcal E}</math>. Thus, outside of <math>{\mathcal E}</math>, the probability that <math>EB</math> is monochromatic is at most <math>5\delta</math>; similarly for <math>AF</math>. Finally, the probability that <math>BF</math> is monochromatic outside of <math>{\mathcal E}</math> is at most <math>7\delta</math>. We conclude that outside of an event of probability
:<math>\frac{1}{2}+5\delta + 3\delta+3\delta+3\delta+5\delta+5\delta+7\delta = \frac{1}{2} + 31\delta,</math>
none of the ten edges connecting <math>A,B,D,E,F</math> are monochromatic. But by the pigeonhole principle, this cannot occur in a 4-coloring, hence <math>\frac{1}{2} + 31 \delta \geq 1</math>, and the first claim follows.

For the second claim, we need to use an iterative argument, by feeding the bounds obtained back into the place in the proof where Lemma 2 is currently invoked. To have all occurring distances stay larger than <math>d</math>, we only need to check <math>|BD| \ge d</math>. Equality occurs when <math>ABD</math> is an equilateral triangle, which means that <math>ABC</math> and <math>ACD</math> are isosceles triangles with sides <math>d,d,1</math> and either with angles <math>150^\circ,15^\circ,15^\circ</math>, or with angles <math>30^\circ,75^\circ,75^\circ</math>. From here calculation gives <math>d \ge \frac{1}{2sin(75^\circ)}=\frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math> and <math>d \le \frac{1}{2sin(15^\circ)}=\frac{\sqrt{3}+1}{\sqrt{2}}=1.9318\dots</math>, but the upper bound is not really important, as for us it is enough that <math>|BD|</math> always stay above <math>d_0=\frac{\sqrt{3}-1}{\sqrt{2}}</math>, which occurs everywhere above this value. Now pick a <math>d\ge d_0</math> for which <math>p_d\ge \frac{1}{2}-\delta-\varepsilon</math>, where <math>\sup_{d\ge d_0} p_d= \frac{1}{2}-\delta</math> and <math>\varepsilon</math> is a small positive number. The calculation of the first case gives <math>\frac{1}{2}+5\delta + 2\delta+2\delta+2\delta+4\delta+4\delta+6\delta+O(\varepsilon) =\frac{1}{2} + 25 \delta+O(\varepsilon) \geq 1</math>, from which <math>\delta\ge 0.02</math> if we choose <math>\varepsilon</math> small enough.

To prove the last claim, we modify the construction; we obtain <math>F</math> by reflecting <math>B</math> to <math>AD</math>, to win <math>2\delta</math> in the last step of the calculation. To invoke Lemma 2, we need (among other things) that <math>EF</math> is at least 1/2, and to iterate in a straight-forward way, we would need a value <math>d_0</math> for which <math>EF</math> is at least <math>d_0</math> if <math>AB</math> is <math>d\ge d_0</math>, but such a <math>d_0</math> doesn't exist. We can, however, still iterate in a weaker sense, as <math>6</math> of the occurring <math>10</math> distances tend to infinity as <math>d=|AB|</math> tends to infinity, and the remaining <math>4</math> are also larger than <math>\frac{\sqrt{3}-1}{\sqrt{2}}</math>, so their probability of them being monochromatic is at most <math>0.48=(0.5-\delta)+(\delta-0.02)</math>. What we get eventually is <math>\frac{1}{2} + 25 \delta-2\delta+ 4(\delta-0.02)+O(\varepsilon) =0.42 + 27 \delta+O(\varepsilon) \geq 1</math>, from which <math>p_d\le \frac{323}{675}=0.4785\ldots</math> for <math>d</math> large enough.
<math>\Box</math>

=== Lemma 37 ===

One has <math>\sup_{0 < d < 2} p_d \geq 1/3</math>.

'''Proof''' For a large integer <math>n</math>, consider the points <math>e^{2\pi i j/n}</math> with <math>j=1,\dots,n</math>. Any unit distance coloring will color these points in at most 3 colors, hence divides the n points into three color classes of some size <math>n_1,n_2,n_3</math>. The number of monochromatic pairs is then
:<math>\frac{n_1(n_1-1)}{2} + \frac{n_2(n_2-1)}{2} + \frac{n_3(n_3-1)}{2} = \frac{1}{2} (n_1^2+n_2^2+n_3^2) + O(n) \geq \frac{1}{6} n^2 + O(n)</math>
by Cauchy-Schwarz. Thus at least <math>1/3-O(1/n)</math> of the pairs are monochromatic. Taking expectations and using the pigeonhole principle, we conclude that one of the distances has a probability at least <math>1/3 -O(1/n)</math> of being monochromatic, giving the claim. <math>\Box</math>

=== Lemma 38 ===

Let ABC be a unit-edge equilateral triangle, and let D be an arbitrary point. Let <math>|AD|, |BD|</math> and <math>|CD|</math> be <math>d,e,f</math> respectively. Then <math>p(d)+p(e)+p(f) \leq 1</math>. In particular, examining the case <math>e=f</math>, if <math>p(d) \geq k</math> then <math>p(\sqrt(d(d \pm \sqrt 3) + 1) \leq (1-k)/2</math>.

'''Proof''' At most one of <math>AD, BD</math> and <math>CD</math> can be monochromatic. <math>\Box</math>

Note: A consequence is that a 4-chromatic unit-distance graph G can demonstrate CNP <math>> 4</math> if, for the {d,e,f} arising from some choice of D above, G contains three equal-sized non-empty sets v_d, v_e, v_f of vertex-pairs such that (a) each vertex-pair within v_d is at distance d (resp. e and f), and (b) in any 4-colouring of G, more than 1/3 of the vertex-pairs in the union of the three sets are monochromatic. Note that this demonstration does not require that v_d contain all the vertex-pairs of G that are at distance d (resp. e and f), nor even that the graph {A,B,C,D} which gives rise to {d,e,f} be a subgraph of G. It seems plausible to find such a graph that is small (and/or symmetrical) enough that its colourings can be human-analysed to establish this property.

=== Lemma 39 ===

If <math> 0 < d \le \frac{\sqrt{6}-\sqrt{2}}{2}=0.517... </math>, then <math> p_d \geq 1/50 </math>.

If <math>\sqrt 2\le d \le 2</math>, then <math>p_d \geq 1/50</math>.

'''Proof''' Take two equilateral unit triangles with one joint vertex, and denote the vertices by <math>A,B,C</math> and <math>C,D,E</math>. Pick the angle of the two triangles such that <math>|AD|=|BE|=d</math>. If <math> 0 < d \le \frac{\sqrt{6}-\sqrt{2}}{2} </math> or <math>\sqrt 2\le d \le 2</math>, then we also have that <math>\frac{\sqrt{3}-1}{\sqrt{2}}\le |AE|,|BD|</math>. Since one of these four non-unit edges must go between monochromatic points, we can apply Proposition 36 to conclude <math>2p_d \geq 1-2\cdot\frac{12}{25}</math>. <math>\Box</math>

Note 1: Slightly larger intervals can be covered (with worse bounds) if we also take the case into account, when only one of <math> AE </math> and <math> BD </math> is at least <math> \frac{\sqrt{3}-1}{\sqrt{2}} </math>, while the other is either in <math>\left[\frac{2}{\sqrt{15}},\frac{\sqrt{3}-1}{\sqrt{2}}\right)</math> or in <math> \left[\frac{1}{2},\frac{2}{\sqrt{15}}\right)</math>. These bounds can be found in the table above.

Note 2: We also can get lower bounds if we swap the 1 and the <math> d </math> segment lengths, but Lemma 40 provides stronger bounds for these cases.

=== Lemma 40 ===

If <math> \frac{1}{2} \le d \le \sqrt{(3-\sqrt{3})\left(1-\frac{1}{\sqrt{2}}\right)}=0.609... </math>, then <math> p_d \geq 1/125 </math>.

If <math> d \ge \sqrt{(3-\sqrt{3})\left(1+\frac{1}{\sqrt{2}}\right)}=1.471... </math>, then <math> p_d \geq 1/125 </math>.

'''Proof''' Take two equilateral triangles with one joint vertex and side-lengths 1 and <math> d </math>, and denote their vertices by <math>A,B,C</math> and <math>C,D,E</math> respectively. Pick the angle of the two triangles such that <math>|AD|=|BE|=d</math>. If <math> \frac{1}{2} \le d \le \sqrt{(3-\sqrt{3})\left(1-\frac{1}{\sqrt{2}}\right)} </math> or <math> \sqrt{(3-\sqrt{3})\left(1+\frac{1}{\sqrt{2}}\right)} \le d </math>, then we also have that <math>\frac{\sqrt{3}-1}{\sqrt{2}}\le |AE|,|BD|</math>. Since one of these seven non-unit edges must go between monochromatic points, we can apply Proposition 36 to conclude <math>5p_d \geq 1-2\cdot\frac{12}{25}</math>. <math>\Box</math>

Note 1: Slightly larger intervals can be covered (with worse bounds) if we also take the case into account, when only one of <math> AE </math> and <math> BD </math> is at least <math> \frac{\sqrt{3}-1}{\sqrt{2}} </math>, while the other is either in <math>\left[\frac{2}{\sqrt{15}},\frac{\sqrt{3}-1}{\sqrt{2}}\right)</math> or in <math> \left[\frac{1}{2},\frac{2}{\sqrt{15}}\right)</math>. These bounds can be found in the table above.

Note 2: We also can get lower bounds if we swap the 1 and the <math> d </math> segment lengths, but Lemma 39 provides stronger bounds for these cases.

== Simplification rules for triplets of points in the complex plane ==
Deduced from the rule <math>{\bf P}(A\land B)+{\bf P}(A\land \lnot B)={\bf P}(A)</math>.
:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) = {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) - {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) ) - {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) \neq {\mathbf c}(z_0) ) + {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) = {\mathbf c}(z_0) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

== Proofs of bounds for conditional probabilities ==
The trivial case, valid where <math>\left|d\right|\neq 1</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) = {\mathbf c}(d) )=0</math>

Trivial plus Baye's Theorem, valid where <math>d\neq 0</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) )=\frac{{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )}\leq 1</math>
Rearrange:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )+{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1</math>

Spindle method: for <math>\left|d\right|=d\geq 1/2</math> and <math>\theta=2\text{arcsin}\left(\frac{1}{2d}\right)</math>:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{i\theta}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) ) = \frac{1}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )} - 1\leq 1</math>
which is another way to see <math>\left|d\right|=d\geq 1/2\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1/2</math>.

== Further questions ==
* For <math>n,m\geq CNP</math>, what consistent relationships exist between <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert n\text{ colors}\right)</math> and <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert m\text{ colors}\right)</math>? How can these relationships be used to sharpen arguments of the probabilistic formulation?

[[Category:Polymath16]]

Hadwiger-Nelson problem

2019-04-16T20:21:37Z

Aubrey: /* Best known results for the chromatic number of spheres */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/08/28/polymath16-tenth-thread-open-sat-instances/ Polymath16, tenth thread: Open SAT instances], Dustin Mixon, Aug 28, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/ Polymath16, eleventh thread: Chromatic numbers of planar sets], Dustin Mixon, Sep 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/ Polymath16, twelfth thread: Year in review and future plans], Dustin Mixon, Mar 23, 2010. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| '''553'''
| 2722
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! Lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

==Best known results for the chromatic number of spheres==

Here we list known results about spheres in the 3-dimensional space, i.e., about 2-dimensional spheres.
The distance is measured in 3-dim and not on the surface!
Most bounds are due to G.J. Simmons. For a nice summary, see [Malen|https://arxiv.org/pdf/1412.2091.pdf].

{| border=1
|-
! Radius !! Lower bound on CN !! comments !! Upper bound on CN !! comments
|-
| <math>r<1/2</math>
| 1
| no unit distances
| 1
| monochromatic sphere
|-
| <math>r=1/2</math>
| 2
| unit distances exist
| 2
| monochromatic hemispheres
|-
| <math>1/2 < r < \sqrt{(3-\sqrt 3)}/2=0.563..</math>
| 3
| odd cycles exist
| 4
| tiling given by facets of tetrahedron
|-
| <math>\sqrt{(3-\sqrt 3)}/2 < r < 1/\sqrt 3</math>
| 3
| odd cycles exist
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 3=0.577..</math>
| 4
|
| 5
| tiling given by facets of square pyramid
|-
| <math>r=1/\sqrt 2=0.707..</math>
| 4
|
| 4
| tiling given by facets of octahedron
|-
| <math>1/\sqrt 3 < r \le \sqrt 3/2=0.866..</math>
| 4
| Moser-spindle
| 6
| tiling given by facets of dodecahedron
|-
| <math>r</math> big generic
| 4
| Moser-spindle
| ?
| Upper bound 8 should be easy, but we want 7
|}

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Probabilistic formulation of Hadwiger-Nelson problem

2018-08-06T14:14:26Z

Aubrey: /* Bounds on p_d for 4-colourings */

Suppose for sake of contradiction that we have a 4-coloring <math>c: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with no unit edges monochromatic, thus

:<math>c(z) \neq c(w) \hbox{ whenever } |z-w| = 1. \quad (1)</math>

We can create further such colorings by composing <math>c</math> on the left with a permutation <math>\sigma \in S_4</math> on the left, and with the (inverse of) a Euclidean isometry <math>T \in E(2)</math> on the right, thus creating a new coloring <math>\sigma \circ c \circ T^{-1}: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with the same property. This is an action of the solvable group <math>S_4 \times E(2)</math>.

It is a fact that all solvable groups (viewed as discrete groups) are [https://en.wikipedia.org/wiki/Amenable_group amenable], so in particular <math>S_4 \times E(2)</math> is amenable. This means that there is a finitely additive probability measure <math>\mu</math> on <math>S_4 \times E(2)</math> (with all subsets of this group measurable), which is left-invariant: <math>\mu(gE) = \mu(E)</math> for all <math>g \in S_4 \times E(2)</math> and <math>E \subset S_4 \times E(2)</math>. This gives <math>S_4 \times E(2)</math> the structure of a finitely additive probability space. We can then define a random coloring <math>{\mathbf c}: {\bf C} \to \{1,2,3,4\}</math> by defining <math>{\mathbf c} := {\mathbf \sigma} \circ c \circ {\mathbf T}^{-1}</math>, where <math>({\mathbf \sigma},{\mathbf T})</math> is the element of the sample space <math>S_4 \times E(2)</math>. Thus for any complex number <math>z</math>, the random color <math>{\mathbf c}(z)</math> is a random variable taking values in <math>\{1,2,3,4\}</math>. The left-invariance of the measure implies that for any <math>(\sigma,T) \in S_4 \times E(2)</math>, the coloring <math> \sigma \circ {\mathbf c} \circ T^{-1}</math> has the same law as <math>{\mathbf c}</math>. This gives the color permutation invariance

:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(z_1) = \sigma(c_1), \dots, {\mathbf c}(z_k) = \sigma(c_k) )\quad (2)</math>

for any <math>z_1,\dots,z_k \in {\bf C}</math>, <math>c_1,\dots,c_k \in \{1,2,3,4\}</math>, and <math>\sigma \in S_4</math>, and the Euclidean isometry invariance
:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(T(z_1)) = c_1, \dots, {\mathbf c}(T(z_k)) = c_k. \quad (3)</math>
(In probabilistic language, this means that the random coloring is a [https://en.wikipedia.org/wiki/Stationary_process stationary process] with respect to the action of <math>S_4 \times E(2)</math>. The extraction of a stationary process from a deterministic object is an example of the ''Furstenberg correspondence principle''.)

== Bounds on <math>p_d</math> for 4-colourings ==

A class of correlations that is of particular interest is that of vertex pairs at some distance <math>d</math>. Accordingly, define
:<math>p_d := {\bf P}( \mathbf{c}(0) = \mathbf{c}(d) ).</math>

{| border=1
|-
! distance !! Lower bound !! Lower-bounding graph/method !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math> \geq 1/2</math>
|
|
| 1/2
| Spindle
|
|-
| <math>\geq 2/\sqrt{15}</math>
|
|
| <math>15/31</math>
| Proposition 36
|
|-
| <math>\geq (\sqrt{3}-1)/\sqrt{2}</math>
|
|
| <math>12/25</math>
| Proposition 36
|
|-
| large enough
|
|
| <math>323/675 = 0.4785\ldots</math>
| Proposition 36
|
|-
| <math>\sqrt{2}\le d \le 2</math>
|<math>1/50</math>
|
|
| Lemma 39
|
|-
| <math> d\geq 1/n, n>1</math>
|
|
| <math>1 - 1/n</math>
| (n+1)-gon with one edge length 1 and the rest d, Lemma 34
|
|-
| <math> d\geq 1/(n \sqrt{3}), n>1</math>
|
|
| <math>(3n-2)/3n</math>
| (n+1)-gon with one edge length <math>1/\sqrt{3}</math> and the rest d, Lemma 34
| Not better than the above on intervals <math>\left(\frac{1}{7},\frac{1}{4\sqrt{3}}\right),\left(\frac{1}{4},\frac{1}{2\sqrt{3}}\right)</math>
|-
| <math>1</math>
| <math>0</math>
| Unit edge
| <math>0</math>
| Unit edge
|
|-
| <math>1/\sqrt{3}</math>
| <math>13/150</math>
| Unit triangle ABC with center O and one vertex at distance <math>1/\sqrt{3}</math> from A,B,O gives <math>6p_{1/\sqrt{3}}+p_{2/\sqrt{3}}\ge 1</math>, upper bound <math>p_{2/\sqrt{3}}\le 12/25</math> finishes proof.
| <math>1/3</math>
| Unit triangle plus its centre
|
|-
| <math>2</math>
| <math>1/4</math>
| <math>G_{21}</math>
| <math>12/25</math>
| Proposition 36
| Lower bound computer verified
|-
| <math>\sqrt{3}</math>
| <math>13/50</math>
| H, Lemma 15+Prop 36
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{7}</math>
|
|
| <math>37/100</math>
| Lemma 38 and Corollary 16
|
|-
| <math>(\sqrt{5}+1)/2</math>
| <math>1/5</math>
| regular pentagon with unit side length
| <math>2/5</math>
| regular pentagon with unit side length
|
|-
| <math>(\sqrt{5}-1)/2</math>
| <math>1/5</math>
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
| <math>2/5</math>
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
|
|-
| <math>\sqrt{11/3}</math>
| <math>1/118</math>
| <math>G_{40}</math> - obsolete
| <math>12/25</math>
| Proposition 36
| computer-verified
|-
| <math>8/3</math>
| <math>1</math>
| <math>V \oplus V \oplus H</math>
| <math>12/25</math>
| Proposition 36
| computer-verified; leads to contradiction
|-
| <math>(\sqrt{6} \pm \sqrt{2})/2</math>
| <math>1/6</math>
| An arrangement of five vertices
| <math>1/2</math> and | <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{2}</math>
| <math>1/14</math>
| A graph of 13 vertices
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math>
| <math>1/28</math>
| A graph of 13 vertices
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{3/2 + \sqrt{33}/6}</math>
| <math>1/196</math>
| A graph of 9 vertices - obsolete
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{5/3}</math>
| <math>1/756</math>
| A graph of 33 vertices - obsolete
| <math>12/25</math>
| Proposition 36
|
|-
| <math>2/\sqrt{3}</math>
| <math>1/177</math>
| A graph of 103 vertices - obsolete
| <math>12/25</math>
| Proposition 36
| Computer-verified
|-
| <math>(\sqrt{33} \pm 1)/(2\sqrt{3})</math>
| <math>1/2</math>
| <math>G_{420}</math>
| <math>12/25</math>
| Proposition 36
| Computer-verified
|}

== Bounds on <math>{\bf P}( \mathbf{c}(0) = \mathbf{c}(d_1) \mid \mathbf{c}(0) \neq \mathbf{c}(d_0) )</math> for 4-colourings ==

{| border=1
|-
! <math>d_1</math> !! <math>d_0</math> !! Lower bound !! Lower-bounding graph !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math>1+(-1)^{1/3}</math>
| <math>2</math>
| 1/2
| H
|
|
| Equals <math>p_{\sqrt 3}/(1-p_2)</math>
|-
| <math>1+(-1)^{1/3}</math>
| <math>1+(-1)^{-1/3}</math>
|
|
| 1/2
| H
|
|}

== Proofs of bounds ==

One can compute some correlations of the coloring exactly:

=== Lemma 1 ===
Let <math>z,w \in {\bf C}</math> with <math>|z-w|=1</math>. Then
:<math> {\bf P}( \mathbf{c}(z) = c ) = \frac{1}{4}\quad (4)</math>
for all <math>c=1,\dots,4</math>,
:<math> {\bf P}( \mathbf{c}(z) = \mathbf{c}(w) ) = 0\quad (5),</math>
and
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c' ) = \frac{1}{12} \quad (6)</math>
for any distinct <math>c,c' \in \{1,2,3,4\}</math>. If <math>u</math> is at a unit distance from both z and w, then
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c'; \mathbf{c}(u) = c'' ) = \frac{1}{24} \quad (6')</math>

'''Proof''' By color invariance (2), the four probabilities in (4) are equal and sum to 1, giving (4). The claim (5) is immediate from (1). From (5) and color invariance, the 12 probabilities in (6) are equal and sum to 1, giving (6). The same argument gives (6').<math>\Box</math>

=== Lemma 2 ===
(Spindle argument) Let <math>|d| \geq 1/2</math>. Then
:<math> p_d \leq \frac{1}{2} \quad (7).</math>

'''Proof''' We can find an angle <math>\theta</math> with <math>|de^{i\theta}-d|=1</math>, then <math>\mathbf{c}(de^{i\theta}) \neq \mathbf{c}(d)</math> almost surely. This means that at least one of the events <math>\mathbf{c}(0) = \mathbf{c}(d)</math>, <math>\mathbf{c}(0) = \mathbf{c}(d e^{i\theta})</math> occurs with probability at most 1/2. The claim now follows from isometry invariance (3). <math>\Box</math>

=== Lemma 3 ===
(Using the K graph) We have
:<math>52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) + {\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} ) \geq 1 \quad (8).</math>

'''Proof''' Consider the 61-vertex graph <math>K</math> from [https://arxiv.org/abs/1804.02385 de Grey's paper]. It has 26 (isometric) copies of H, and thus 52 copies of the triangle <math>(1, e^{2\pi i/3}, e^{4\pi i/3})</math>. With probability at least <math>1 - 52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) </math>, none of these triangles are monochromatic. By the argument in that paper, this implies that the three linking diagonals <math>(-2, +2)</math>, <math>(-2 e^{2\pi i/3}, 2e^{2\pi i/3})</math>, <math>(-2 e^{4\pi i/3}, e^{-4\pi i/3})</math> are monochromatic. This gives the claim. <math>\Box</math>

=== Corollary 4 ===
(Existence of monochromatic <math>\sqrt{3}</math>-triangles) We have <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) \geq \frac{1}{104}</math>.

'''Proof''' The probability <math>{\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} )</math> is at most <math>{\bf P}( \mathbf{c}(-2) = \mathbf{c}(2)) = p_4</math>, which by Lemma 2 is at most 1/2. The claim now follows from Lemma 3. <math>\Box</math>

=== Computer-verified Claim 5 ===
(Using the graph M) One has <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = 0</math> (Note this contradicts Corollary 4).

'''Proof''' This simply reflects the fact that there is no 4-coloring of the 1345-vertex graph M from [https://arxiv.org/abs/1804.02385 de Grey's paper] with its central copy of H containing a monochromatic triangle. One can use other graphs for this purpose, such as the 278-vertex graph <math>M_1</math> or <math>V \oplus V \oplus H</math>. <math>\Box</math>

=== Computer-verified Claim 6 ===
(Using the graph <math>V \oplus V \oplus H</math>) One has <math>p_{8/3} = 1</math> (note this contradicts Lemma 2).

'''Proof''' This reflects the fact that every 4-coloring of <math>V \oplus V \oplus H</math> must assign the same color to 0 and 8/3. There is also a 745-vertex subgraph of <math>V \oplus V \oplus H</math> with the same property. <math>\Box</math>

=== Lemma 7 ===
(Using <math>G_{40}</math>) We have
:<math>59 p_{\sqrt{11/3}} + p_{8/3} \geq 1</math>.

'''Proof''' This reflects the fact that every 4-coloring of the 40-vertex graph <math>G_{40}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which none of the 59 pairs of vertices at distance <math>\sqrt{11/3}</math> apart, will assign the same color to 0 and 8/3. (This is presumably human-verifiable.) <math>\Box</math>

=== Corollary 8 ===
One has
:<math> p_{\sqrt{11/3}} \geq \frac{1}{118}</math>.

'''Proof''' Combine Lemma 7 and Lemma 2. <math>\Box</math>

=== Lemma 9 ===
(Using <math>G_{49}</math>) One has
:<math>18 {\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq p_{\sqrt{11/3}} .</math>

'''Proof''' This reflects the fact that every 4-coloring of the 49-vertex graph <math>G_{49}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which 0 and <math>\sqrt{11/3}</math> have the same color, at least one of the 18 copies of <math>(1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3)</math> is monochromatic. This is potentially human-verifiable. <math>\Box</math>

=== Corollary 10 ===
(Existence of monochromatic <math>1/\sqrt{3}</math>-triangles) One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq \frac{1}{2124}.</math>

'''Proof''' Combine Corollary 8 and Lemma 9. <math>\Box</math>

=== Computer-verified claim 11 ===
One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) = 0</math>. (This contradicts Corollary 10).

'''Proof''' This reflects the fact that the 627-vertex graph <math>G_{627}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] does not have any 4-colorings with <math>1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3</math> monochromatic. <math>\Box</math>

=== Lemma 12 ===
For certain special distances d, one can improve the bound in Lemma 2:

If <math>k \geq 1</math> is a natural number, <math>j\in\mathbb{Z}</math>, <math>\gcd(j,2k+1)=1</math>, and <math>r = \frac{1}{2} \csc\left(\frac{j\pi}{2k+1}\right)</math> then
:<math> p_r \leq \frac{k}{2k+1},</math>
thus for instance
:<math> p_{\frac{1}{\sqrt{3}}} \leq \frac{1}{3}.</math>

'''Proof''' Observe that the regular 2k+1-polygon <math>r, re^{2\pi i/(2k+1)}, r e^{4\pi i/(2k+1)}, \dots, r^{4k\pi i/(k+1)}</math> has unit side lengths. By the pigeonhole principle, we conclude that at most k of these vertices can have the same color as the origin, and the claim follows. <math>\Box</math>

On the other hand, for <math>k=2,j=1</math> we also know from the regular pentagon of unit sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}+1}{2}} \leq \frac{2}{5} \quad (9)</math>
since any 4-coloring of that pentagon has either one or two monochromatic diagonals.

Similarly, for <math>k=2,j=2</math> we also know from the regular pentagon of <math>\frac{\sqrt{5}-1}{2}</math> sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}-1}{2}} \leq \frac{2}{5}</math>
since any 4-coloring of that pentagon has either one or two monochromatic edges. More generally, if <math>a,b,c,d,e</math> are the diagonal lengths of a pentagon with unit sides, then
:<math> 1 \leq p_a + p_b + p_c + p_d + p_e \leq 2.</math>

=== Lemma 13 ===
We have
:<math> 7 p_{\frac{1}{\sqrt{3}}} \geq p_{\sqrt{3}}</math>.

'''Proof''' Consider the unit rhombus <math>0, 1, e^{i\pi/3}, e^{-i\pi/3}</math> together with the centers <math>e^{i\pi/6}/\sqrt{3}, e^{-i\pi/6}/\sqrt{3}</math>. With probability <math>p_{\sqrt{3}}</math>, the two far vertices <math>e^{i\pi/3}, e^{-i\pi/3}</math> are the same color, and then 0,1 will be two other colors. This forces either one of the centers <math>e^{i\pi/6}/\sqrt{3}</math> of a triangle to have a common color with one of the vertices of that triangle, or the two centers must have the same color. Thus in any event one of the seven edges of distance <math>1/\sqrt{3}</math> is monochromatic, giving the claim. <math>\Box</math>

=== Corollary 14 ===
We have <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{728}</math>.

This slightly improves upon the lower bound of 1/2124 coming from Corollary 10.

'''Proof''' Combine Corollary 4 and Lemma 13. <math>\Box</math>

=== Lemma 15 ===
One has
:<math>p_{\sqrt{3}} + p_2 \geq \frac{2}{3}</math> and <math>2 p_{\sqrt{3}} + p_2 \geq 1</math>.

'''Proof''' As noted in de Grey's paper, there are essentially four 4-colorings of H. H has six edges of length <math>\sqrt{3}</math> and three of length <math>2</math>. If we let a denote the number of monochromatic <math>\sqrt{3}</math> edges and b the number of monochromatic <math>2</math>-edges, we see from inspection of all four colorings that <math>(a,b)</math> is either <math>(6, 0), (4,0), (2, 1)</math>, or <math>(0,3)</math>. In particular, one always has <math>\frac{a}{6} + \frac{b}{3} \geq \frac{2}{3}</math> and <math>2\frac{a}{6} + \frac{b}{3} \geq 1</math>. Taking expectations, we obtain the claim. <math>\Box</math>

=== Corollary 16 ===
We have <math>p_2 \geq \frac{1}{6}</math>, <math>p_{\sqrt{3}} \geq \frac{1}{4} </math> and <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{28}</math>.

'''Proof''' Combine Lemma 2, Lemma 15, and Lemma 13. <math>\Box</math>

=== Lemma 17 ===
If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths a,b,c. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also, thus
:<math>{\bf P}(\mathbf{c}(0) \neq \mathbf{c}(a)) + {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(b)) \geq {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(c))</math>
and the claim follows. <math>\Box</math>

Note that Lemma 2 follows from the a=b, c=1 case of this lemma. Iterating this lemma starting with Lemma 2 we can also obtain slightly nontrivial upper bounds on <math>p_a</math> for small values of a, e.g. <math>p_a \leq 1 - 2^{-k}</math> when <math>a \geq 2^{-k}, k\in\mathbb{Z}^+</math>.

Further, we can generalise the a=b case to one in which the triangle is replaced by a (k+1)-gon of which one edge is 1 and the others are all equal, leading to the stronger result <math>p_a \leq 1 - 1/k</math> when <math>a \geq 1/k, k\in\mathbb{Z}^+ \land k>1</math>. Further strengthening is achieved by using <math>1/\sqrt{3}</math> as the long edge, given Lemma 12.

=== Lemma 18 ===
Whenever <math>d>0</math>, one has the inequalities
:<math> |p_{\phi d} - p_d| \leq \frac{2}{5}, p_{\phi d} + p_d \geq \frac{1}{5}, 2p_d - p_{\phi d} \leq 1, 2 p_{\phi d} - p_d \leq 1</math>
where <math>\phi := \frac{1+\sqrt{5}}{2}</math> is the golden ratio. Also we have
:<math> p_{d/\sqrt{3}} \leq \frac{1}{3} + p_d, \frac{1}{2} + \frac{1}{2} p_d.</math>

Note that this generalises (9), as well as a special case of Lemma 12.

'''Proof''' Consider the regular pentagon with sidelength <math>d</math>, so it also has 5 diagonals of length <math>\phi d</math>. Let <math>a \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic edges and let <math>b \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic diagonals. Observe:
* <math>a,b</math> cannot both be zero (pigeonhole principle).
* <math>a</math> cannot be 4. Similarly, <math>b</math> cannot be 4.
* If <math>a=5</math>, then <math>b=5</math>, and conversely.
* If <math>a=0</math>, then <math>b=1,2</math>; similarly, if <math>b=0</math>, then <math>a=1,2</math>.

From this we observe the inequalities
:<math> |\frac{a}{5}-\frac{b}{5}| \leq \frac{2}{5}; \frac{a}{5} + \frac{b}{5} \geq \frac{1}{5}; 2 \frac{a}{5} - \frac{b}{5} \leq 1; 2\frac{b}{5} - \frac{a}{5} \leq 1</math>
and on taking expectations we obtain the first claim. Similarly, if one considers the colorings of an equilateral triangle of sidelength <math>d</math> together with its center, and counts the numbers <math>a,b \in \{0,1,2,3\}</math> of monochromatic edges of length <math>d</math> and <math>d/\sqrt{3}</math> respectively, one observes that one always has <math>\frac{b}{3} \leq \frac{1}{3} + \frac{2}{3} \frac{a}{3}, \frac{1}{2} + \frac{1}{2} \frac{a}{3}</math>, and on taking expectations one obtains the claim.<math>\Box</math>

The hexagon <math>H</math> has essentially four distinct colorings: the coloring <math>\hbox{2tri}</math> with two triangles, the coloring <math>\hbox{1tri}</math> with one triangle, the coloring <math>\hbox{axisym}</math> that is symmetric around an axis, and the coloring <math>\hbox{centralsym}</math> that is symmetric around the central point. This gives four probabilities <math>p_{H = 2tri}, p_{H = 1tri}, p_{H = axisym}, p_{H = centralsym}</math> that sum to 1. By counting the number of monochromatic edges of length <math>\sqrt{3}, 2</math> respectively, one also obtains the identities
:<math>p_{\sqrt{3}} = p_{H = 2tri} + \frac{2}{3} p_{H = 1tri} + \frac{1}{3} p_{H = axisym}; \quad p_2 = \frac{1}{3} p_{H=axisym} + p_{H=centralsym}</math>
which reproves Lemma 15. Also
:<math> {\bf P}( \mathbf{c}(0) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = p_{H = 2tri} + \frac{1}{2} p_{H=1tri}.</math>
Any 4-coloring of L contains at least one triangle within one of its 52 copies of H, thus
:<math> p_{H = 2tri} + \frac{1}{2} p_{H=1tri} \geq \frac{1}{52}</math>
which reproves Corollary 4.

=== Lemma 19 ===
(Hubai) One has <math>p_{H = 1tri} + p_{H = axisym} \geq \frac{1}{10}</math>.

'''Proof''' Consider five copies of H centred at 0,1,2,3,4. With probability at least <math>1 - 5( p_{H = 1tri} + p_{H = axisym} )</math>, none of these copies of H are colored 1tri or axisym, and so must be colored 2tri or centralsym. One can check then that if one of the copies is colored 2tri, then so is any adjacent copy; thus all five copies are colored 2tri, or all five are colored centralsym. In either case we see that -1 and 5 are colored the same color. Comparing with Lemma 2 then gives the claim. <math>\Box</math>

=== Theorem 20 ===
One has <math>p_{H = 1tri} > 0</math>.

'''Proof''' Suppose for contradiction that <math>p_{H = 1tri} = 0</math>. One can then run a version of the de Bruijn-Erdos argument to obtain a coloring in which 1tri hexagons are completely nonexistent (since there are arbitrarily large finite colorings with this property). Consider the triangular lattice <math>{\bf Z}[e^{\pi i/3}]</math>. We 2-color the edges of this lattice by coloring an edge black if it is the short diagonal of a unit rhombus with monochromatic long diagonal, and white otherwise. The four colorings of hexagons lead to four possible colorings at each vertex:

* If H is colored 2tri, then all six edges to the centre of H are black.
* If H is colored 1tri, then two edges to the centre of H at 120 degree angles are white, the other four are black.
* If H is colored axisym, then two opposing edges of the centre of H are black, the other four are white.
* If H is colored centralsym, then all six edges to the centre of H are black.

In particular, as we are assuming no 1tri hexagons, the faces cut out by the black edges have angles 60 degrees, and thus must be equilateral triangles, sectors of angle 60, half-planes, or the entire plane. If there is at least one equilateral triangle, then the rest of the black edges must form an equilateral lattice with that triangle sidelength. This leads to only a small number of possible hexagon colorings in the lattice:

# Case 1: All edges white.
# Case 2: All edges black.
# Case 3.k: For some natural number <math>k \geq 2</math>, the length k edges joining adjacent vertices in some coset of <math>k \cdot {\mathbf Z}[ e^{\pi i/3} ]</math> are all black, and the remaining edges are white.
# Case 4: Each horizontal row consists either entirely of black edges, or entirely of white edges.
# Case 5: Each northwest row consists either entirely of black edges, or entirely of white edges.
# Case 6: Each northeast row consists either entirely of black edges, or entirely of white edges.
# Case 7: Six rays of black edges meeting at a common vertex; all other edges white.

Technically, Case 1 is contained in Cases 4,5,6 as written above, but this will not be an issue. One can view Case 7 as a limiting case <math>k=\infty</math> of Case 3.k; Case 2 is similarly the opposite limiting case <math>k=1</math>.

In the first case, the coloring is periodic with periods <math>2, 2 e^{\pi i/3}</math>. In the second case, it is periodic with periods <math>3, 3 e^{\pi i/3}</math>. In the third case, it is periodic with periods <math>3k, 3k e^{\pi i/3}</math>. Also note that for each k, one can check if Case 3.k holds by inspecting the coloring at a finite number of vertices. Thus the event that Case 3.k holds is "measurable" in the sense that a meaningful probability can be assigned. (But Cases 1,2,4,5,6 are not measurable events, they require an infinite number of points to be inspected, and the probability measure we are using is only finitely additive rather than infinitely additive.) In Case 4, the coloring is periodic with period 2; also, every coset of <math>2 \cdot {\bf Z}[e^{\pi i/3}]</math> is 2-colored. Similarly for Case 5 and 6 (where the periods are <math>2 e^{2\pi i/3}</math> and <math>2 e^{4\pi i/3}</math> respectively.)

Let <math>\alpha_k</math> be the probability that Case 3.k holds for the given value of <math>k</math>. Then <math> \sum_{k=2}^K \alpha_k \leq 1</math> for any k, hence <math>\sum_{k=2}^\infty \alpha_k \leq 1</math>. In particular, we can find <math>K_1</math> such that <math>\sum_{k={K_1}}^\infty \alpha_k \leq 0.1</math> (say). Let <math>P</math> be six times the least common multiple of <math>1,2,\dots,K_1</math>. Then the coloring is P- and <math>P e^{\pi i/3}</math>-periodic for Case 1, Case 2, and all Case 3.k with <math>k \leq K_1</math>. On the other hand, if <math>K_2</math> is sufficiently large depending on <math>P</math>, and Case 3.k holds for some <math>k \geq K_2</math>, then almost all of the hexagons are colored centralsym, which makes the coloring "almost <math>P, P e^{\pi i/3}</math>-periodic" in the sense that
:<math>{\bf c}(z+P e^{\pi i j/3}) = {\bf c}(z) \hbox{ for } j=0,1,2,3,4,5</math>
will hold for at least <math>0.9</math> of the lattice points <math>z \in {\bf Z}[e^{\pi i/3}]</math> with <math>|z| \leq K_2</math>. Similarly for Case 7 (which is sort of a <math>k=\infty</math> limiting case of Case 3.k.) Thus, with the probability <math> \geq 1 - \sum_{k=K_1}^{K_2} \alpha_k \geq 0.9</math>, the coloring of the seven vertices <math>{\bf c}(0), {\bf c}(P e^{\pi ij/3}, j=1,\dots,6</math> is (up to rotation and recoloring) one of the three patterns of the central and linking vertices in Figure 3 of Aubrey's paper, namely

* <math>{\bf c}(0) = {\bf c}(P) = {\bf c}(P e^{\pi i/3}) = {\bf c}(P e^{2\pi i/3}) = {\bf c}(P e^{3\pi i/3}) = {\bf c}(P e^{4\pi i/3}) = {\bf c}(P e^{5\pi i/3}) </math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{3\pi i/3})</math>; <math>{\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{3\pi i/3})</math>; <math> {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>.

Using the spindling argument from Aubrey's paper, we conclude that the third possibility must in fact hold with probability at least 0.8; on the other hand, from Lemma 2 this scenario can only occur with probability at most 1/2, giving the required contradiction.
<math>\Box</math>

One should be able to refine this argument to show that <math>p_{H = 1tri} > c</math> for an absolute constant <math> c>0 </math>.

=== Lemma 21 ===
Providing a tighter bound for Lemma 17 with a more thorough proof: If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths <math>a,b,c</math>. Let <math>\left|z_2\right|=b,\left|a-z_2\right|=c</math>. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also: <math>\mathbf{c}(a)\neq\mathbf{c}(z_2)\Rightarrow[\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)]</math>. <math>[A\Rightarrow B]\Rightarrow {\bf P}(A)\leq{\bf P}(B)</math> thus
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) \geq {\bf P}(\mathbf{c}(a) \neq \mathbf{c}(z_2)) = 1-p_c</math>
<math>{\bf P}(A\lor B) +{\bf P}(A\land B)={\bf P}(A)+{\bf P}(B)</math>, so
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)) + {\bf P}(\mathbf{c}(0)\neq\mathbf{c}(z_2)) - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>1-p_c \leq 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
Using the law of cosines: <math>z_2=b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)</math>
:<math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math> <math>\Box</math>

=== Lemma 22 ===

One has <math>3 p_{1/\sqrt{3}} \geq {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) )</math>.

'''Proof''' Let <math>z</math> be a complex number of magnitude <math>1/\sqrt{3}</math> that is a unit distance from 1. If <math>\mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) = c</math> (say), then <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> cannot be colored with <math>c</math>; also, <math>z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> are the vertices of a unit equilateral triangle and thus must take on three different colors. By the pigeonhole principle, one of <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> must then take the same color as the origin, and the claim follows. <math>\Box</math>

=== Lemma 23 ===

One has <math>4 p_{(\sqrt{6} \pm \sqrt{2})/2} + p_{\sqrt{2}} \geq 1</math>. In particular, by Lemma 2, <math>p_{(\sqrt{6}+\sqrt{2})/2} \geq 1/8</math>.

'''Proof''' [ExIs2018b] We just prove the claim for the + sign (the - sign can then be obtained after applying the Galois conjugacy that maps <math>\sqrt{3}</math> to <math>-\sqrt{3}</math>, leaving <math>\sqrt{2}</math> unchanged). Set <math>d := \frac{\sqrt{6}+\sqrt{2}}{2}</math>, and consider the five vertices

:<math>0, e^{5\pi i/4}, e^{5\pi i/4} + d, e^{5\pi i/4} + e^{\pi i/3} d, e^{5\pi i/4} + (e^{\pi i/3}-i)d.</math>

One can check that of the ten edges determined by these five vertices, five have unit length, four have length d, and the remaining distance (from 0 to <math>e^{5\pi i/4}+d</math>) has distance <math>\sqrt{2}</math>. Since <math>\mathbf{c}</math> must make one of the latter five edges monochromatic, the claim follows. <math>\Box</math>

=== Lemma 24 ===

One has <math>p_{\sqrt{2}} \geq \frac{1}{14}</math>.

'''Proof''' In page 7 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 20 unit distance edges and 14 edges of length <math>\sqrt{2}</math> is constructed. The coloring <math>\mathbf{c}</math> must make one of the 14 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 25 ===

Let <math>d = \frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math> and <math>e = \frac{3^{1/4} \sqrt{2} + \sqrt{3} - 1}{2}</math>.
Then one has <math>14 p_d + p_e \geq 1</math>. In particular, by Lemma 2, <math>p_d \geq 1/28</math>.

'''Proof''' In page 9 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 19 unit edges, 14 edges of length d, and one edge of length e is constructed. The coloring <math>\mathbf{c}</math> must make one of the 15 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 26 ===

Let <math>d = \sqrt{3/2 + \sqrt{33}/6}</math>. Then <math>7 p_d \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_d \geq \frac{1}{196}</math>.

'''Proof''' In page 11 of [ExIs2018b], a graph of nine vertices consisting of 12 unit edges and 7 edges of length d is constructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Thus, <math>\mathbf{c}</math> can only make the AB edge monochromatic if one of the seven length d edges is monochromatic. The claim follows.
<math>\Box</math>

=== Lemma 27 ===

One has <math>27 p_{\sqrt{5/3}} \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_{\sqrt{5/3}} \geq \frac{1}{756}</math>.

'''Proof''' In page 13 of [ExIs2018], a graph of 33 vertices with some unit edges and 27 edges of length <math>\sqrt{5/3}</math> is contructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Now repeat the proof of Lemma 26. <math>\Box</math>

=== Computer-verified claim 28 ===

One has <math>p_{2/\sqrt{3}} \geq \frac{1}{177}</math>.

'''Proof''' In page 15 of [ExIs2018], a 5-chromatic graph of 103 vertices, 312 unit edges, and 177 edges of length <math>2/\sqrt{3}</math> is constructed. <math>\mathbf{c}</math> must make one of the latter edges monochromatic, giving the claim. <math>\Box</math>

=== Lemma 29 ===

One has <math>p_{(\sqrt{6} \pm \sqrt{2})/2} \geq 1/6</math> (this improves the bound in Lemma 23).

'''Proof''' Use graphs 505 and 507 from [S2004] and the spindle bound. <math>\Box</math>

=== Lemma 30 ===
For <math>m > n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' <math>n</math> colors and <math>m</math> points necessitates at least 2 having equal color. I.e.

:<math>{\bf P}\left(\bigvee_{k=0}^n \bigvee_{j=k+1}^n\ \mathbf{c}(z_k) = \mathbf{c}(z_j)\right) = 1</math>

The lemma then follows immediately from the fact:
:<math>{\bf P}\left(\bigcup_{k} E_k\right) \leq \sum_{k} {\bf P}\left(E_k\right) \,\Box</math>

=== Corollary 31 ===
Let <math>\lvert z_k\rvert=1</math>. Then for <math>m \geq n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' Use lemma 30 on the set <math>\left\{z_k \bigg\vert 1\leq k\leq m \land k\in\mathbb{Z}\right\}\cup\{0\}</math>. Simplify using <math>{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(0) \right)=0</math>.
<math>\Box</math>

=== Lemma 32 ===
For <math>x\in\mathbb{R}</math> and an <math>n</math>-coloring of the plane, <math>\sum_{k=1}^{n-1}\left(n-k\right){\bf P}\left(\mathbf{c}\left(0\right) = \mathbf{c}\left( 2\sin\left(\frac{kx}{2}\right) \right) \right) \geq 1</math>.

'''Proof''' Use corollary 31 on the set <math>\left\{e^{ikx} \bigg\vert 0\leq k < n \land k\in\mathbb{Z}\right\}</math>. and simplify by grouping lengths.
<math>\Box</math>

=== Corollary 33 ===
Interesting(easy to simplify results of) values for <math>x</math> in Lemma 32 are in <math>\left\{x \bigg\vert \sin\left(\frac{kx}{2}\right)=1 \land 1\leq k < n \land k\in\mathbb{Z}\right\}</math>.
For 4-colorings, this gives
:<math>2p_{\sqrt 3}+p_2 \geq 1</math>
:<math>3p_{(\sqrt 3-1)/\sqrt 2}+p_{\sqrt 2} \geq 1</math>
:<math>3p_{2\sin(\pi/18)}+2p_{2\sin(\pi/9)} \geq 1</math>

=== Lemma 34 ===
Generalizing the note of Lemma 17, <math>\lvert d_1\rvert= d_1 > \lvert d_0\rvert= d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>

'''Proof''' let <math>\lvert z_{j+1} -z_j\rvert=d_0 > 0, \lvert z_{j+n} -z_0\rvert=d_1>0</math>.

Base case, <math>n=2</math>, by Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle:
:<math>2d_0\geq d_1\Rightarrow 2p_{d_0}\leq 1+p_{d_1}</math>
The inductive step is Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle. After induction:
:<math>[n\geq 2\land nd_0\geq d_1]\Rightarrow np_{d_0}\leq n-1+p_{d_1}</math>
Substitute <math>n=\left\lceil\frac{d_1}{d_0}\right\rceil</math>, simplify, rearrange:
:<math>d_1 > d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>
<math>\Box</math>

=== Proposition 35 ===

If <math>d > 1/\sqrt{2}</math> obeys the inequalities
:<math>\sin( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
and
:<math>\cos( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
then
:<math>p_d \leq \frac{1}{2} - \frac{1}{188}.</math>

(One can check that the conditions are obeyed precisely when <math>d \geq \frac{\sqrt{33}-1}{8} = 0.84307\dots</math>.)

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. Since <math>AB</math> is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the triangle <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>.

Now let <math>BADE</math> be a rhombus with sidelengths d and <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. By the hypotheses, the diagonals BD, AE of this rhombus have length at least 1/2, and hence are monochromatic with probability at most 1/2 by Lemma 2. As above, ABD and BDE are each monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>. As BD is monochromatic with probability at most <math>1/2</math>, we conclude that BADE is monochormatic with probability at least <math>\frac{1}{2}-6\delta</math>.

Now let <math>EDFG</math> be another rhombus congruent to <math>BADE</math>. As BD, AE have length at least 1/2, at least one of the long diagonals BF, AG have length at least 1/2 (the diagonal opposite an obtuse or right-angled triangle will work). Let's say BF has length at least 1/2. As BADE and EDFG are both monochromatic with probability at least <math>\frac{1}{2}-6\delta</math>, and the common edge DE is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the entire configuration ABDEFG is monochromatic with probability at least <math>\frac{1}{2}-11\delta</math>. In particular the pentagon ABDEF is monochromatic with at least this probability. However, in this pentagon, the five edges BA, AD, DE, EB, EF are monochromatic with probability at most <math>\frac{1}{2}-\delta</math>, and the other five edges are monochromatic with probability at most <math>\frac{1}{2}</math> by Lemma 2. Thus the probability that at least one of the edges of this pentagon is monochromatic is at most <math>(\frac{1}{2}-11\delta) + 5 \times 10\delta + 5 \times 11\delta = \frac{1}{2}+94\delta</math>. On the other hand, by the pigeonhole principle, this probability is 1. The claim follows. <math>\Box</math>

=== Proposition 36 ===

If <math>d \ge \frac{2}{\sqrt{15}} = 0.5163\dots</math>, then <math>p_d \leq \frac{1}{2} - \frac{1}{62}</math>,

if <math>d \ge \frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math>, then <math>p_d \leq 0.48</math>,

and <math>\limsup_d p_d\leq \frac{323}{675}=0.4785\ldots</math> (so <math>p_d<0.4786</math> if <math>d</math> is large enough).

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. A simple calculation shows that if <math>d \ge \frac{2}{\sqrt{15}}</math>, then <math>|BD| \ge \frac{1}{2}</math>.
If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. By inclusion-exclusion, we conclude that outside of the event that <math>AB</math> is monochromatic, the probability that <math>AD</math> is monochromatic is at most <math>2\delta</math>.

Now let <math>ADE</math> the <math>180^\circ</math> rotation of <math>ADB</math> around the midpoint of <math>AD</math>, and let <math>FDE</math> be the <math>180^\circ</math> rotation of <math>ADE</math> around the midpoint of <math>DE</math>. By the hypotheses, the line segments <math>AE, BD, BE, BF, DF</math> all have length at least 1/2. Let <math>{\mathcal E}</math> be the event that at least one of <math>AB, AD, DE, EF</math> is monochromatic. By the previous paragraph, this event occurs with probability at most <math>\frac{1}{2}-\delta+2\delta+2\delta+2\delta = \frac{1}{2}+5\delta</math>.

By previous considerations, <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>, and this event lies in <math>{\mathcal E}</math>. On the other hand, <math>BD</math> is monochromatic with probability at most 1/2 by Lemma 2. We conclude that outside of <math>{\mathcal E}</math>, <math>BD</math> is only monochromatic with probability at most <math>3\delta</math>. A similar argument (replacing <math>ABD</math> by <math>DAE</math> or <math>EDF</math>) shows that outside of <math>{\mathcal E}</math>, <math>AE</math> is monochromatic with probability at most <math>3\delta</math>, and similarly for <math>DF</math>. Now we consider <math>EB</math>. By previous considerations, the probability that <math>ABDE</math> is monochromatic is at least <math>\frac{1}{2}-5\delta</math>, and this event lies inside <math>{\mathcal E}</math>. Thus, outside of <math>{\mathcal E}</math>, the probability that <math>EB</math> is monochromatic is at most <math>5\delta</math>; similarly for <math>AF</math>. Finally, the probability that <math>BF</math> is monochromatic outside of <math>{\mathcal E}</math> is at most <math>7\delta</math>. We conclude that outside of an event of probability
:<math>\frac{1}{2}+5\delta + 3\delta+3\delta+3\delta+5\delta+5\delta+7\delta = \frac{1}{2} + 31\delta,</math>
none of the ten edges connecting <math>A,B,D,E,F</math> are monochromatic. But by the pigeonhole principle, this cannot occur in a 4-coloring, hence <math>\frac{1}{2} + 31 \delta \geq 1</math>, and the first claim follows.

For the second claim, we need to use an iterative argument, by feeding the bounds obtained back into the place in the proof where Lemma 2 is currently invoked. To have all occurring distances stay larger than <math>d</math>, we only need to check <math>|BD| \ge d</math>. Equality occurs when <math>ABD</math> is an equilateral triangle, which means that <math>ABC</math> and <math>ACD</math> are isosceles triangles with sides <math>d,d,1</math> and either with angles <math>150^\circ,15^\circ,15^\circ</math>, or with angles <math>30^\circ,75^\circ,75^\circ</math>. From here calculation gives <math>d \ge \frac{1}{2sin(75^\circ)}=\frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math> and <math>d \le \frac{1}{2sin(15^\circ)}=\frac{\sqrt{3}+1}{\sqrt{2}}=1.9318\dots</math>, but the upper bound is not really important, as for us it is enough that <math>|BD|</math> always stay above <math>d_0=\frac{\sqrt{3}-1}{\sqrt{2}}</math>, which occurs everywhere above this value. Now pick a <math>d\ge d_0</math> for which <math>p_d\ge \frac{1}{2}-\delta-\varepsilon</math>, where <math>\sup_{d\ge d_0} p_d= \frac{1}{2}-\delta</math> and <math>\varepsilon</math> is a small positive number. The calculation of the first case gives <math>\frac{1}{2}+5\delta + 2\delta+2\delta+2\delta+4\delta+4\delta+6\delta+O(\varepsilon) =\frac{1}{2} + 25 \delta+O(\varepsilon) \geq 1</math>, from which <math>\delta\ge 0.02</math> if we choose <math>\varepsilon</math> small enough.

To prove the last claim, we modify the construction; we obtain <math>F</math> by reflecting <math>B</math> to <math>AD</math>, to win <math>2\delta</math> in the last step of the calculation. To invoke Lemma 2, we need (among other things) that <math>EF</math> is at least 1/2, and to iterate in a straight-forward way, we would need a value <math>d_0</math> for which <math>EF</math> is at least <math>d_0</math> if <math>AB</math> is <math>d\ge d_0</math>, but such a <math>d_0</math> doesn't exist. We can, however, still iterate in a weaker sense, as <math>6</math> of the occurring <math>10</math> distances tend to infinity as <math>d=|AB|</math> tends to infinity, and the remaining <math>4</math> are also larger than <math>\frac{\sqrt{3}-1}{\sqrt{2}}</math>, so their probability of them being monochromatic is at most <math>0.48=(0.5-\delta)+(\delta-0.02)</math>. What we get eventually is <math>\frac{1}{2} + 25 \delta-2\delta+ 4(\delta-0.02)+O(\varepsilon) =0.42 + 27 \delta+O(\varepsilon) \geq 1</math>, from which <math>p_d\le \frac{323}{675}=0.4785\ldots</math> for <math>d</math> large enough.
<math>\Box</math>

=== Lemma 37 ===

One has <math>\sup_{0 < d < 2} p_d \geq 1/3</math>.

'''Proof''' For a large integer <math>n</math>, consider the points <math>e^{2\pi i j/n}</math> with <math>j=1,\dots,n</math>. Any unit distance coloring will color these points in at most 3 colors, hence divides the n points into three color classes of some size <math>n_1,n_2,n_3</math>. The number of monochromatic pairs is then
:<math>\frac{n_1(n_1-1)}{2} + \frac{n_2(n_2-1)}{2} + \frac{n_3(n_3-1)}{2} = \frac{1}{2} (n_1^2+n_2^2+n_3^2) + O(n) \geq \frac{1}{6} n^2 + O(n)</math>
by Cauchy-Schwarz. Thus at least <math>1/3-O(1/n)</math> of the pairs are monochromatic. Taking expectations and using the pigeonhole principle, we conclude that one of the distances has a probability at least <math>1/3 -O(1/n)</math> of being monochromatic, giving the claim. <math>\Box</math>

=== Lemma 38 ===

Let ABC be a unit-edge equilateral triangle, and let D be an arbitrary point. Let <math>|AD|, |BD|</math> and <math>|CD|</math> be <math>d,e,f</math> respectively. Then <math>p(d)+p(e)+p(f) \leq 1</math>. In particular, examining the case <math>e=f</math>, if <math>p(d) \geq k</math> then <math>p(\sqrt(d(d \pm \sqrt 3) + 1) \leq (1-k)/2</math>.

'''Proof''' At most one of <math>AD, BD</math> and <math>CD</math> can be monochromatic. <math>\Box</math>

Note: A consequence is that a 4-chromatic unit-distance graph G can demonstrate CNP <math>> 4</math> if, for the {d,e,f} arising from some choice of D above, G contains three equal-sized non-empty sets v_d, v_e, v_f of vertex-pairs such that (a) each vertex-pair within v_d is at distance d (resp. e and f), and (b) in any 4-colouring of G, more than 1/3 of the vertex-pairs in the union of the three sets are monochromatic. Note that this demonstration does not require that v_d contain all the vertex-pairs of G that are at distance d (resp. e and f), nor even that the graph {A,B,C,D} which gives rise to {d,e,f} be a subgraph of G. It seems plausible to find such a graph that is small (and/or symmetrical) enough that its colourings can be human-analysed to establish this property.

=== Lemma 39 ===

If <math>\sqrt 2\le d \le 2</math>, then <math>p_d \geq 1/50</math>.

'''Proof''' Take two equilateral unit triangles that have one joint vertex, and denote the vertices by <math>A,B,C</math> and <math>C,D,E</math>. Pick the angle of the two triangles such that <math>|AD|=|BE|=d</math>. If <math>\sqrt 2\le d \le 2</math>, then we also have that <math>\frac{\sqrt{3}-1}{\sqrt{2}}\le |AE|,|BD|</math>. Since one of these four non-unit edges must go between monochromatic points, we can apply Proposition 36 to conclude <math>2p_d \geq 1-2\cdot\frac{12}{25}</math>. <math>\Box</math>

== Simplification rules for triplets of points in the complex plane ==
Deduced from the rule <math>{\bf P}(A\land B)+{\bf P}(A\land \lnot B)={\bf P}(A)</math>.
:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) = {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) - {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) ) - {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) \neq {\mathbf c}(z_0) ) + {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) = {\mathbf c}(z_0) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

== Proofs of bounds for conditional probabilities ==
The trivial case, valid where <math>\left|d\right|\neq 1</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) = {\mathbf c}(d) )=0</math>

Trivial plus Baye's Theorem, valid where <math>d\neq 0</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) )=\frac{{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )}\leq 1</math>
Rearrange:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )+{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1</math>

Spindle method: for <math>\left|d\right|=d\geq 1/2</math> and <math>\theta=2\text{arcsin}\left(\frac{1}{2d}\right)</math>:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{i\theta}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) ) = \frac{1}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )} - 1\leq 1</math>
which is another way to see <math>\left|d\right|=d\geq 1/2\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1/2</math>.

== Further questions ==
* For <math>n,m\geq CNP</math>, what consistent relationships exist between <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert n\text{ colors}\right)</math> and <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert m\text{ colors}\right)</math>? How can these relationships be used to sharpen arguments of the probabilistic formulation?

[[Category:Polymath16]]

Probabilistic formulation of Hadwiger-Nelson problem

2018-08-01T18:56:27Z

Aubrey: /* Bounds on p_d for 4-colourings */

Suppose for sake of contradiction that we have a 4-coloring <math>c: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with no unit edges monochromatic, thus

:<math>c(z) \neq c(w) \hbox{ whenever } |z-w| = 1. \quad (1)</math>

We can create further such colorings by composing <math>c</math> on the left with a permutation <math>\sigma \in S_4</math> on the left, and with the (inverse of) a Euclidean isometry <math>T \in E(2)</math> on the right, thus creating a new coloring <math>\sigma \circ c \circ T^{-1}: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with the same property. This is an action of the solvable group <math>S_4 \times E(2)</math>.

It is a fact that all solvable groups (viewed as discrete groups) are [https://en.wikipedia.org/wiki/Amenable_group amenable], so in particular <math>S_4 \times E(2)</math> is amenable. This means that there is a finitely additive probability measure <math>\mu</math> on <math>S_4 \times E(2)</math> (with all subsets of this group measurable), which is left-invariant: <math>\mu(gE) = \mu(E)</math> for all <math>g \in S_4 \times E(2)</math> and <math>E \subset S_4 \times E(2)</math>. This gives <math>S_4 \times E(2)</math> the structure of a finitely additive probability space. We can then define a random coloring <math>{\mathbf c}: {\bf C} \to \{1,2,3,4\}</math> by defining <math>{\mathbf c} := {\mathbf \sigma} \circ c \circ {\mathbf T}^{-1}</math>, where <math>({\mathbf \sigma},{\mathbf T})</math> is the element of the sample space <math>S_4 \times E(2)</math>. Thus for any complex number <math>z</math>, the random color <math>{\mathbf c}(z)</math> is a random variable taking values in <math>\{1,2,3,4\}</math>. The left-invariance of the measure implies that for any <math>(\sigma,T) \in S_4 \times E(2)</math>, the coloring <math> \sigma \circ {\mathbf c} \circ T^{-1}</math> has the same law as <math>{\mathbf c}</math>. This gives the color permutation invariance

:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(z_1) = \sigma(c_1), \dots, {\mathbf c}(z_k) = \sigma(c_k) )\quad (2)</math>

for any <math>z_1,\dots,z_k \in {\bf C}</math>, <math>c_1,\dots,c_k \in \{1,2,3,4\}</math>, and <math>\sigma \in S_4</math>, and the Euclidean isometry invariance
:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(T(z_1)) = c_1, \dots, {\mathbf c}(T(z_k)) = c_k. \quad (3)</math>
(In probabilistic language, this means that the random coloring is a [https://en.wikipedia.org/wiki/Stationary_process stationary process] with respect to the action of <math>S_4 \times E(2)</math>. The extraction of a stationary process from a deterministic object is an example of the ''Furstenberg correspondence principle''.)

== Bounds on <math>p_d</math> for 4-colourings ==

A class of correlations that is of particular interest is that of vertex pairs at some distance <math>d</math>. Accordingly, define
:<math>p_d := {\bf P}( \mathbf{c}(0) = \mathbf{c}(d) ).</math>

{| border=1
|-
! distance !! Lower bound !! Lower-bounding graph/method !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math> \geq 1/2</math>
|
|
| 1/2
| Spindle
|
|-
| <math>\geq 2/\sqrt{15}</math>
|
|
| <math>15/31</math>
| Proposition 36
|
|-
| <math>\geq (\sqrt{3}-1)/\sqrt{2}</math>
|
|
| <math>12/25</math>
| Proposition 36
|
|-
| large enough
|
|
| <math>323/675 = 0.4785\ldots</math>
| Proposition 36
|
|-
| <math> d\geq 1/n, n>1</math>
|
|
| <math>1 - 1/n</math>
| (n+1)-gon with one edge length 1 and the rest d, Lemma 34
|
|-
| <math> d\geq 1/(n \sqrt{3}), n>1</math>
|
|
| <math>(3n-2)/3n</math>
| (n+1)-gon with one edge length <math>1/\sqrt{3}</math> and the rest d, Lemma 34
| Not better than the above on intervals <math>\left(\frac{1}{7},\frac{1}{4\sqrt{3}}\right),\left(\frac{1}{4},\frac{1}{2\sqrt{3}}\right)</math>
|-
| <math>1</math>
| <math>0</math>
| Unit edge
| <math>0</math>
| Unit edge
|
|-
| <math>1/\sqrt{3}</math>
| <math>1/28</math>
| Unit diamond plus centres of triangles, together with H, Corollary 16
| <math>1/3</math>
| Unit triangle plus its centre
|
|-
| <math>2</math>
| <math>1/4</math>
| <math>G_{21}</math>
| <math>12/25</math>
| Proposition 36
| Lower bound computer verified
|-
| <math>\sqrt{3}</math>
| <math>13/50</math>
| H, Prop 36
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{7}</math>
|
|
| <math>37/100</math>
| Lemma 38 and Corollary 16
|
|-
| <math>(\sqrt{5}+1)/2</math>
| <math>1/5</math>
| regular pentagon with unit side length
| <math>2/5</math>
| regular pentagon with unit side length
|
|-
| <math>(\sqrt{5}-1)/2</math>
| <math>1/5</math>
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
| <math>2/5</math>
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
|
|-
| <math>\sqrt{11/3}</math>
| <math>1/118</math>
| <math>G_{40}</math>
| <math>12/25</math>
| Proposition 36
| computer-verified
|-
| <math>8/3</math>
| <math>1</math>
| <math>V \oplus V \oplus H</math>
| <math>12/25</math>
| Proposition 36
| computer-verified; leads to contradiction
|-
| <math>(\sqrt{6} \pm \sqrt{2})/2</math>
| <math>1/6</math>
| An arrangement of five vertices
| <math>1/2</math> and | <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{2}</math>
| <math>1/14</math>
| A graph of 13 vertices
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math>
| <math>1/28</math>
| A graph of 13 vertices
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{3/2 + \sqrt{33}/6}</math>
| <math>1/196</math>
| A graph of 9 vertices
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{5/3}</math>
| <math>1/756</math>
| A graph of 33 vertices
| <math>12/25</math>
| Proposition 36
|
|-
| <math>2/\sqrt{3}</math>
| <math>1/177</math>
| A graph of 103 vertices
| <math>12/25</math>
| Proposition 36
| Computer-verified
|-
| <math>(\sqrt{33} \pm 1)/(2\sqrt{3})</math>
| <math>1/2</math>
| <math>G_{420}</math>
| <math>12/25</math>
| Proposition 36
| Computer-verified
|}

== Bounds on <math>{\bf P}( \mathbf{c}(0) = \mathbf{c}(d_1) \mid \mathbf{c}(0) \neq \mathbf{c}(d_0) )</math> for 4-colourings ==

{| border=1
|-
! <math>d_1</math> !! <math>d_0</math> !! Lower bound !! Lower-bounding graph !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math>1+(-1)^{1/3}</math>
| <math>2</math>
| 1/2
| H
|
|
| Equals <math>p_{\sqrt 3}/(1-p_2)</math>
|-
| <math>1+(-1)^{1/3}</math>
| <math>1+(-1)^{-1/3}</math>
|
|
| 1/2
| H
|
|}

== Proofs of bounds ==

One can compute some correlations of the coloring exactly:

=== Lemma 1 ===
Let <math>z,w \in {\bf C}</math> with <math>|z-w|=1</math>. Then
:<math> {\bf P}( \mathbf{c}(z) = c ) = \frac{1}{4}\quad (4)</math>
for all <math>c=1,\dots,4</math>,
:<math> {\bf P}( \mathbf{c}(z) = \mathbf{c}(w) ) = 0\quad (5),</math>
and
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c' ) = \frac{1}{12} \quad (6)</math>
for any distinct <math>c,c' \in \{1,2,3,4\}</math>. If <math>u</math> is at a unit distance from both z and w, then
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c'; \mathbf{c}(u) = c'' ) = \frac{1}{24} \quad (6')</math>

'''Proof''' By color invariance (2), the four probabilities in (4) are equal and sum to 1, giving (4). The claim (5) is immediate from (1). From (5) and color invariance, the 12 probabilities in (6) are equal and sum to 1, giving (6). The same argument gives (6').<math>\Box</math>

=== Lemma 2 ===
(Spindle argument) Let <math>|d| \geq 1/2</math>. Then
:<math> p_d \leq \frac{1}{2} \quad (7).</math>

'''Proof''' We can find an angle <math>\theta</math> with <math>|de^{i\theta}-d|=1</math>, then <math>\mathbf{c}(de^{i\theta}) \neq \mathbf{c}(d)</math> almost surely. This means that at least one of the events <math>\mathbf{c}(0) = \mathbf{c}(d)</math>, <math>\mathbf{c}(0) = \mathbf{c}(d e^{i\theta})</math> occurs with probability at most 1/2. The claim now follows from isometry invariance (3). <math>\Box</math>

=== Lemma 3 ===
(Using the K graph) We have
:<math>52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) + {\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} ) \geq 1 \quad (8).</math>

'''Proof''' Consider the 61-vertex graph <math>K</math> from [https://arxiv.org/abs/1804.02385 de Grey's paper]. It has 26 (isometric) copies of H, and thus 52 copies of the triangle <math>(1, e^{2\pi i/3}, e^{4\pi i/3})</math>. With probability at least <math>1 - 52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) </math>, none of these triangles are monochromatic. By the argument in that paper, this implies that the three linking diagonals <math>(-2, +2)</math>, <math>(-2 e^{2\pi i/3}, 2e^{2\pi i/3})</math>, <math>(-2 e^{4\pi i/3}, e^{-4\pi i/3})</math> are monochromatic. This gives the claim. <math>\Box</math>

=== Corollary 4 ===
(Existence of monochromatic <math>\sqrt{3}</math>-triangles) We have <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) \geq \frac{1}{104}</math>.

'''Proof''' The probability <math>{\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} )</math> is at most <math>{\bf P}( \mathbf{c}(-2) = \mathbf{c}(2)) = p_4</math>, which by Lemma 2 is at most 1/2. The claim now follows from Lemma 3. <math>\Box</math>

=== Computer-verified Claim 5 ===
(Using the graph M) One has <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = 0</math> (Note this contradicts Corollary 4).

'''Proof''' This simply reflects the fact that there is no 4-coloring of the 1345-vertex graph M from [https://arxiv.org/abs/1804.02385 de Grey's paper] with its central copy of H containing a monochromatic triangle. One can use other graphs for this purpose, such as the 278-vertex graph <math>M_1</math> or <math>V \oplus V \oplus H</math>. <math>\Box</math>

=== Computer-verified Claim 6 ===
(Using the graph <math>V \oplus V \oplus H</math>) One has <math>p_{8/3} = 1</math> (note this contradicts Lemma 2).

'''Proof''' This reflects the fact that every 4-coloring of <math>V \oplus V \oplus H</math> must assign the same color to 0 and 8/3. There is also a 745-vertex subgraph of <math>V \oplus V \oplus H</math> with the same property. <math>\Box</math>

=== Lemma 7 ===
(Using <math>G_{40}</math>) We have
:<math>59 p_{\sqrt{11/3}} + p_{8/3} \geq 1</math>.

'''Proof''' This reflects the fact that every 4-coloring of the 40-vertex graph <math>G_{40}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which none of the 59 pairs of vertices at distance <math>\sqrt{11/3}</math> apart, will assign the same color to 0 and 8/3. (This is presumably human-verifiable.) <math>\Box</math>

=== Corollary 8 ===
One has
:<math> p_{\sqrt{11/3}} \geq \frac{1}{118}</math>.

'''Proof''' Combine Lemma 7 and Lemma 2. <math>\Box</math>

=== Lemma 9 ===
(Using <math>G_{49}</math>) One has
:<math>18 {\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq p_{\sqrt{11/3}} .</math>

'''Proof''' This reflects the fact that every 4-coloring of the 49-vertex graph <math>G_{49}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which 0 and <math>\sqrt{11/3}</math> have the same color, at least one of the 18 copies of <math>(1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3)</math> is monochromatic. This is potentially human-verifiable. <math>\Box</math>

=== Corollary 10 ===
(Existence of monochromatic <math>1/\sqrt{3}</math>-triangles) One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq \frac{1}{2124}.</math>

'''Proof''' Combine Corollary 8 and Lemma 9. <math>\Box</math>

=== Computer-verified claim 11 ===
One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) = 0</math>. (This contradicts Corollary 10).

'''Proof''' This reflects the fact that the 627-vertex graph <math>G_{627}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] does not have any 4-colorings with <math>1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3</math> monochromatic. <math>\Box</math>

=== Lemma 12 ===
For certain special distances d, one can improve the bound in Lemma 2:

If <math>k \geq 1</math> is a natural number, <math>j\in\mathbb{Z}</math>, <math>\gcd(j,2k+1)=1</math>, and <math>r = \frac{1}{2} \csc\left(\frac{j\pi}{2k+1}\right)</math> then
:<math> p_r \leq \frac{k}{2k+1},</math>
thus for instance
:<math> p_{\frac{1}{\sqrt{3}}} \leq \frac{1}{3}.</math>

'''Proof''' Observe that the regular 2k+1-polygon <math>r, re^{2\pi i/(2k+1)}, r e^{4\pi i/(2k+1)}, \dots, r^{4k\pi i/(k+1)}</math> has unit side lengths. By the pigeonhole principle, we conclude that at most k of these vertices can have the same color as the origin, and the claim follows. <math>\Box</math>

On the other hand, for <math>k=2,j=1</math> we also know from the regular pentagon of unit sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}+1}{2}} \leq \frac{2}{5} \quad (9)</math>
since any 4-coloring of that pentagon has either one or two monochromatic diagonals.

Similarly, for <math>k=2,j=2</math> we also know from the regular pentagon of <math>\frac{\sqrt{5}-1}{2}</math> sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}-1}{2}} \leq \frac{2}{5}</math>
since any 4-coloring of that pentagon has either one or two monochromatic edges. More generally, if <math>a,b,c,d,e</math> are the diagonal lengths of a pentagon with unit sides, then
:<math> 1 \leq p_a + p_b + p_c + p_d + p_e \leq 2.</math>

=== Lemma 13 ===
We have
:<math> 7 p_{\frac{1}{\sqrt{3}}} \geq p_{\sqrt{3}}</math>.

'''Proof''' Consider the unit rhombus <math>0, 1, e^{i\pi/3}, e^{-i\pi/3}</math> together with the centers <math>e^{i\pi/6}/\sqrt{3}, e^{-i\pi/6}/\sqrt{3}</math>. With probability <math>p_{\sqrt{3}}</math>, the two far vertices <math>e^{i\pi/3}, e^{-i\pi/3}</math> are the same color, and then 0,1 will be two other colors. This forces either one of the centers <math>e^{i\pi/6}/\sqrt{3}</math> of a triangle to have a common color with one of the vertices of that triangle, or the two centers must have the same color. Thus in any event one of the seven edges of distance <math>1/\sqrt{3}</math> is monochromatic, giving the claim. <math>\Box</math>

=== Corollary 14 ===
We have <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{728}</math>.

This slightly improves upon the lower bound of 1/2124 coming from Corollary 10.

'''Proof''' Combine Corollary 4 and Lemma 13. <math>\Box</math>

=== Lemma 15 ===
One has
:<math>p_{\sqrt{3}} + p_2 \geq \frac{2}{3}</math> and <math>2 p_{\sqrt{3}} + p_2 \geq 1</math>.

'''Proof''' As noted in de Grey's paper, there are essentially four 4-colorings of H. H has six edges of length <math>\sqrt{3}</math> and three of length <math>2</math>. If we let a denote the number of monochromatic <math>\sqrt{3}</math> edges and b the number of monochromatic <math>2</math>-edges, we see from inspection of all four colorings that <math>(a,b)</math> is either <math>(6, 0), (4,0), (2, 1)</math>, or <math>(0,3)</math>. In particular, one always has <math>\frac{a}{6} + \frac{b}{3} \geq \frac{2}{3}</math> and <math>2\frac{a}{6} + \frac{b}{3} \geq 1</math>. Taking expectations, we obtain the claim. <math>\Box</math>

=== Corollary 16 ===
We have <math>p_2 \geq \frac{1}{6}</math>, <math>p_{\sqrt{3}} \geq \frac{1}{4} </math> and <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{28}</math>.

'''Proof''' Combine Lemma 2, Lemma 15, and Lemma 13. <math>\Box</math>

=== Lemma 17 ===
If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths a,b,c. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also, thus
:<math>{\bf P}(\mathbf{c}(0) \neq \mathbf{c}(a)) + {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(b)) \geq {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(c))</math>
and the claim follows. <math>\Box</math>

Note that Lemma 2 follows from the a=b, c=1 case of this lemma. Iterating this lemma starting with Lemma 2 we can also obtain slightly nontrivial upper bounds on <math>p_a</math> for small values of a, e.g. <math>p_a \leq 1 - 2^{-k}</math> when <math>a \geq 2^{-k}, k\in\mathbb{Z}^+</math>.

Further, we can generalise the a=b case to one in which the triangle is replaced by a (k+1)-gon of which one edge is 1 and the others are all equal, leading to the stronger result <math>p_a \leq 1 - 1/k</math> when <math>a \geq 1/k, k\in\mathbb{Z}^+ \land k>1</math>. Further strengthening is achieved by using <math>1/\sqrt{3}</math> as the long edge, given Lemma 12.

=== Lemma 18 ===
Whenever <math>d>0</math>, one has the inequalities
:<math> |p_{\phi d} - p_d| \leq \frac{2}{5}, p_{\phi d} + p_d \geq \frac{1}{5}, 2p_d - p_{\phi d} \leq 1, 2 p_{\phi d} - p_d \leq 1</math>
where <math>\phi := \frac{1+\sqrt{5}}{2}</math> is the golden ratio. Also we have
:<math> p_{d/\sqrt{3}} \leq \frac{1}{3} + p_d, \frac{1}{2} + \frac{1}{2} p_d.</math>

Note that this generalises (9), as well as a special case of Lemma 12.

'''Proof''' Consider the regular pentagon with sidelength <math>d</math>, so it also has 5 diagonals of length <math>\phi d</math>. Let <math>a \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic edges and let <math>b \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic diagonals. Observe:
* <math>a,b</math> cannot both be zero (pigeonhole principle).
* <math>a</math> cannot be 4. Similarly, <math>b</math> cannot be 4.
* If <math>a=5</math>, then <math>b=5</math>, and conversely.
* If <math>a=0</math>, then <math>b=1,2</math>; similarly, if <math>b=0</math>, then <math>a=1,2</math>.

From this we observe the inequalities
:<math> |\frac{a}{5}-\frac{b}{5}| \leq \frac{2}{5}; \frac{a}{5} + \frac{b}{5} \geq \frac{1}{5}; 2 \frac{a}{5} - \frac{b}{5} \leq 1; 2\frac{b}{5} - \frac{a}{5} \leq 1</math>
and on taking expectations we obtain the first claim. Similarly, if one considers the colorings of an equilateral triangle of sidelength <math>d</math> together with its center, and counts the numbers <math>a,b \in \{0,1,2,3\}</math> of monochromatic edges of length <math>d</math> and <math>d/\sqrt{3}</math> respectively, one observes that one always has <math>\frac{b}{3} \leq \frac{1}{3} + \frac{2}{3} \frac{a}{3}, \frac{1}{2} + \frac{1}{2} \frac{a}{3}</math>, and on taking expectations one obtains the claim.<math>\Box</math>

The hexagon <math>H</math> has essentially four distinct colorings: the coloring <math>\hbox{2tri}</math> with two triangles, the coloring <math>\hbox{1tri}</math> with one triangle, the coloring <math>\hbox{axisym}</math> that is symmetric around an axis, and the coloring <math>\hbox{centralsym}</math> that is symmetric around the central point. This gives four probabilities <math>p_{H = 2tri}, p_{H = 1tri}, p_{H = axisym}, p_{H = centralsym}</math> that sum to 1. By counting the number of monochromatic edges of length <math>\sqrt{3}, 2</math> respectively, one also obtains the identities
:<math>p_{\sqrt{3}} = p_{H = 2tri} + \frac{2}{3} p_{H = 1tri} + \frac{1}{3} p_{H = axisym}; \quad p_2 = \frac{1}{3} p_{H=axisym} + p_{H=centralsym}</math>
which reproves Lemma 15. Also
:<math> {\bf P}( \mathbf{c}(0) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = p_{H = 2tri} + \frac{1}{2} p_{H=1tri}.</math>
Any 4-coloring of L contains at least one triangle within one of its 52 copies of H, thus
:<math> p_{H = 2tri} + \frac{1}{2} p_{H=1tri} \geq \frac{1}{52}</math>
which reproves Corollary 4.

=== Lemma 19 ===
(Hubai) One has <math>p_{H = 1tri} + p_{H = axisym} \geq \frac{1}{10}</math>.

'''Proof''' Consider five copies of H centred at 0,1,2,3,4. With probability at least <math>1 - 5( p_{H = 1tri} + p_{H = axisym} )</math>, none of these copies of H are colored 1tri or axisym, and so must be colored 2tri or centralsym. One can check then that if one of the copies is colored 2tri, then so is any adjacent copy; thus all five copies are colored 2tri, or all five are colored centralsym. In either case we see that -1 and 5 are colored the same color. Comparing with Lemma 2 then gives the claim. <math>\Box</math>

=== Theorem 20 ===
One has <math>p_{H = 1tri} > 0</math>.

'''Proof''' Suppose for contradiction that <math>p_{H = 1tri} = 0</math>. One can then run a version of the de Bruijn-Erdos argument to obtain a coloring in which 1tri hexagons are completely nonexistent (since there are arbitrarily large finite colorings with this property). Consider the triangular lattice <math>{\bf Z}[e^{\pi i/3}]</math>. We 2-color the edges of this lattice by coloring an edge black if it is the short diagonal of a unit rhombus with monochromatic long diagonal, and white otherwise. The four colorings of hexagons lead to four possible colorings at each vertex:

* If H is colored 2tri, then all six edges to the centre of H are black.
* If H is colored 1tri, then two edges to the centre of H at 120 degree angles are white, the other four are black.
* If H is colored axisym, then two opposing edges of the centre of H are black, the other four are white.
* If H is colored centralsym, then all six edges to the centre of H are black.

In particular, as we are assuming no 1tri hexagons, the faces cut out by the black edges have angles 60 degrees, and thus must be equilateral triangles, sectors of angle 60, half-planes, or the entire plane. If there is at least one equilateral triangle, then the rest of the black edges must form an equilateral lattice with that triangle sidelength. This leads to only a small number of possible hexagon colorings in the lattice:

# Case 1: All edges white.
# Case 2: All edges black.
# Case 3.k: For some natural number <math>k \geq 2</math>, the length k edges joining adjacent vertices in some coset of <math>k \cdot {\mathbf Z}[ e^{\pi i/3} ]</math> are all black, and the remaining edges are white.
# Case 4: Each horizontal row consists either entirely of black edges, or entirely of white edges.
# Case 5: Each northwest row consists either entirely of black edges, or entirely of white edges.
# Case 6: Each northeast row consists either entirely of black edges, or entirely of white edges.
# Case 7: Six rays of black edges meeting at a common vertex; all other edges white.

Technically, Case 1 is contained in Cases 4,5,6 as written above, but this will not be an issue. One can view Case 7 as a limiting case <math>k=\infty</math> of Case 3.k; Case 2 is similarly the opposite limiting case <math>k=1</math>.

In the first case, the coloring is periodic with periods <math>2, 2 e^{\pi i/3}</math>. In the second case, it is periodic with periods <math>3, 3 e^{\pi i/3}</math>. In the third case, it is periodic with periods <math>3k, 3k e^{\pi i/3}</math>. Also note that for each k, one can check if Case 3.k holds by inspecting the coloring at a finite number of vertices. Thus the event that Case 3.k holds is "measurable" in the sense that a meaningful probability can be assigned. (But Cases 1,2,4,5,6 are not measurable events, they require an infinite number of points to be inspected, and the probability measure we are using is only finitely additive rather than infinitely additive.) In Case 4, the coloring is periodic with period 2; also, every coset of <math>2 \cdot {\bf Z}[e^{\pi i/3}]</math> is 2-colored. Similarly for Case 5 and 6 (where the periods are <math>2 e^{2\pi i/3}</math> and <math>2 e^{4\pi i/3}</math> respectively.)

Let <math>\alpha_k</math> be the probability that Case 3.k holds for the given value of <math>k</math>. Then <math> \sum_{k=2}^K \alpha_k \leq 1</math> for any k, hence <math>\sum_{k=2}^\infty \alpha_k \leq 1</math>. In particular, we can find <math>K_1</math> such that <math>\sum_{k={K_1}}^\infty \alpha_k \leq 0.1</math> (say). Let <math>P</math> be six times the least common multiple of <math>1,2,\dots,K_1</math>. Then the coloring is P- and <math>P e^{\pi i/3}</math>-periodic for Case 1, Case 2, and all Case 3.k with <math>k \leq K_1</math>. On the other hand, if <math>K_2</math> is sufficiently large depending on <math>P</math>, and Case 3.k holds for some <math>k \geq K_2</math>, then almost all of the hexagons are colored centralsym, which makes the coloring "almost <math>P, P e^{\pi i/3}</math>-periodic" in the sense that
:<math>{\bf c}(z+P e^{\pi i j/3}) = {\bf c}(z) \hbox{ for } j=0,1,2,3,4,5</math>
will hold for at least <math>0.9</math> of the lattice points <math>z \in {\bf Z}[e^{\pi i/3}]</math> with <math>|z| \leq K_2</math>. Similarly for Case 7 (which is sort of a <math>k=\infty</math> limiting case of Case 3.k.) Thus, with the probability <math> \geq 1 - \sum_{k=K_1}^{K_2} \alpha_k \geq 0.9</math>, the coloring of the seven vertices <math>{\bf c}(0), {\bf c}(P e^{\pi ij/3}, j=1,\dots,6</math> is (up to rotation and recoloring) one of the three patterns of the central and linking vertices in Figure 3 of Aubrey's paper, namely

* <math>{\bf c}(0) = {\bf c}(P) = {\bf c}(P e^{\pi i/3}) = {\bf c}(P e^{2\pi i/3}) = {\bf c}(P e^{3\pi i/3}) = {\bf c}(P e^{4\pi i/3}) = {\bf c}(P e^{5\pi i/3}) </math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{3\pi i/3})</math>; <math>{\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{3\pi i/3})</math>; <math> {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>.

Using the spindling argument from Aubrey's paper, we conclude that the third possibility must in fact hold with probability at least 0.8; on the other hand, from Lemma 2 this scenario can only occur with probability at most 1/2, giving the required contradiction.
<math>\Box</math>

One should be able to refine this argument to show that <math>p_{H = 1tri} > c</math> for an absolute constant <math> c>0 </math>.

=== Lemma 21 ===
Providing a tighter bound for Lemma 17 with a more thorough proof: If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths <math>a,b,c</math>. Let <math>\left|z_2\right|=b,\left|a-z_2\right|=c</math>. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also: <math>\mathbf{c}(a)\neq\mathbf{c}(z_2)\Rightarrow[\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)]</math>. <math>[A\Rightarrow B]\Rightarrow {\bf P}(A)\leq{\bf P}(B)</math> thus
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) \geq {\bf P}(\mathbf{c}(a) \neq \mathbf{c}(z_2)) = 1-p_c</math>
<math>{\bf P}(A\lor B) +{\bf P}(A\land B)={\bf P}(A)+{\bf P}(B)</math>, so
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)) + {\bf P}(\mathbf{c}(0)\neq\mathbf{c}(z_2)) - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>1-p_c \leq 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
Using the law of cosines: <math>z_2=b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)</math>
:<math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math> <math>\Box</math>

=== Lemma 22 ===

One has <math>3 p_{1/\sqrt{3}} \geq {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) )</math>.

'''Proof''' Let <math>z</math> be a complex number of magnitude <math>1/\sqrt{3}</math> that is a unit distance from 1. If <math>\mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) = c</math> (say), then <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> cannot be colored with <math>c</math>; also, <math>z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> are the vertices of a unit equilateral triangle and thus must take on three different colors. By the pigeonhole principle, one of <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> must then take the same color as the origin, and the claim follows. <math>\Box</math>

=== Lemma 23 ===

One has <math>4 p_{(\sqrt{6} \pm \sqrt{2})/2} + p_{\sqrt{2}} \geq 1</math>. In particular, by Lemma 2, <math>p_{(\sqrt{6}+\sqrt{2})/2} \geq 1/8</math>.

'''Proof''' [ExIs2018b] We just prove the claim for the + sign (the - sign can then be obtained after applying the Galois conjugacy that maps <math>\sqrt{3}</math> to <math>-\sqrt{3}</math>, leaving <math>\sqrt{2}</math> unchanged). Set <math>d := \frac{\sqrt{6}+\sqrt{2}}{2}</math>, and consider the five vertices

:<math>0, e^{5\pi i/4}, e^{5\pi i/4} + d, e^{5\pi i/4} + e^{\pi i/3} d, e^{5\pi i/4} + (e^{\pi i/3}-i)d.</math>

One can check that of the ten edges determined by these five vertices, five have unit length, four have length d, and the remaining distance (from 0 to <math>e^{5\pi i/4}+d</math>) has distance <math>\sqrt{2}</math>. Since <math>\mathbf{c}</math> must make one of the latter five edges monochromatic, the claim follows. <math>\Box</math>

=== Lemma 24 ===

One has <math>p_{\sqrt{2}} \geq \frac{1}{14}</math>.

'''Proof''' In page 7 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 20 unit distance edges and 14 edges of length <math>\sqrt{2}</math> is constructed. The coloring <math>\mathbf{c}</math> must make one of the 14 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 25 ===

Let <math>d = \frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math> and <math>e = \frac{3^{1/4} \sqrt{2} + \sqrt{3} - 1}{2}</math>.
Then one has <math>14 p_d + p_e \geq 1</math>. In particular, by Lemma 2, <math>p_d \geq 1/28</math>.

'''Proof''' In page 9 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 19 unit edges, 14 edges of length d, and one edge of length e is constructed. The coloring <math>\mathbf{c}</math> must make one of the 15 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 26 ===

Let <math>d = \sqrt{3/2 + \sqrt{33}/6}</math>. Then <math>7 p_d \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_d \geq \frac{1}{196}</math>.

'''Proof''' In page 11 of [ExIs2018b], a graph of nine vertices consisting of 12 unit edges and 7 edges of length d is constructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Thus, <math>\mathbf{c}</math> can only make the AB edge monochromatic if one of the seven length d edges is monochromatic. The claim follows.
<math>\Box</math>

=== Lemma 27 ===

One has <math>27 p_{\sqrt{5/3}} \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_{\sqrt{5/3}} \geq \frac{1}{756}</math>.

'''Proof''' In page 13 of [ExIs2018], a graph of 33 vertices with some unit edges and 27 edges of length <math>\sqrt{5/3}</math> is contructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Now repeat the proof of Lemma 26. <math>\Box</math>

=== Computer-verified claim 28 ===

One has <math>p_{2/\sqrt{3}} \geq \frac{1}{177}</math>.

'''Proof''' In page 15 of [ExIs2018], a 5-chromatic graph of 103 vertices, 312 unit edges, and 177 edges of length <math>2/\sqrt{3}</math> is constructed. <math>\mathbf{c}</math> must make one of the latter edges monochromatic, giving the claim. <math>\Box</math>

=== Lemma 29 ===

One has <math>p_{(\sqrt{6} \pm \sqrt{2})/2} \geq 1/6</math> (this improves the bound in Lemma 23).

'''Proof''' Use graphs 505 and 507 from [S2004] and the spindle bound. <math>\Box</math>

=== Lemma 30 ===
For <math>m > n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' <math>n</math> colors and <math>m</math> points necessitates at least 2 having equal color. I.e.

:<math>{\bf P}\left(\bigvee_{k=0}^n \bigvee_{j=k+1}^n\ \mathbf{c}(z_k) = \mathbf{c}(z_j)\right) = 1</math>

The lemma then follows immediately from the fact:
:<math>{\bf P}\left(\bigcup_{k} E_k\right) \leq \sum_{k} {\bf P}\left(E_k\right) \,\Box</math>

=== Corollary 31 ===
Let <math>\lvert z_k\rvert=1</math>. Then for <math>m \geq n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' Use lemma 30 on the set <math>\left\{z_k \bigg\vert 1\leq k\leq m \land k\in\mathbb{Z}\right\}\cup\{0\}</math>. Simplify using <math>{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(0) \right)=0</math>.
<math>\Box</math>

=== Lemma 32 ===
For <math>x\in\mathbb{R}</math> and an <math>n</math>-coloring of the plane, <math>\sum_{k=1}^{n-1}\left(n-k\right){\bf P}\left(\mathbf{c}\left(0\right) = \mathbf{c}\left( 2\sin\left(\frac{kx}{2}\right) \right) \right) \geq 1</math>.

'''Proof''' Use corollary 31 on the set <math>\left\{e^{ikx} \bigg\vert 0\leq k < n \land k\in\mathbb{Z}\right\}</math>. and simplify by grouping lengths.
<math>\Box</math>

=== Corollary 33 ===
Interesting(easy to simplify results of) values for <math>x</math> in Lemma 32 are in <math>\left\{x \bigg\vert \sin\left(\frac{kx}{2}\right)=1 \land 1\leq k < n \land k\in\mathbb{Z}\right\}</math>.
For 4-colorings, this gives
:<math>2p_{\sqrt 3}+p_2 \geq 1</math>
:<math>3p_{(\sqrt 3-1)/\sqrt 2}+p_{\sqrt 2} \geq 1</math>
:<math>3p_{2\sin(\pi/18)}+2p_{2\sin(\pi/9)} \geq 1</math>

=== Lemma 34 ===
Generalizing the note of Lemma 17, <math>\lvert d_1\rvert= d_1 > \lvert d_0\rvert= d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>

'''Proof''' let <math>\lvert z_{j+1} -z_j\rvert=d_0 > 0, \lvert z_{j+n} -z_0\rvert=d_1>0</math>.

Base case, <math>n=2</math>, by Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle:
:<math>2d_0\geq d_1\Rightarrow 2p_{d_0}\leq 1+p_{d_1}</math>
The inductive step is Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle. After induction:
:<math>[n\geq 2\land nd_0\geq d_1]\Rightarrow np_{d_0}\leq n-1+p_{d_1}</math>
Substitute <math>n=\left\lceil\frac{d_1}{d_0}\right\rceil</math>, simplify, rearrange:
:<math>d_1 > d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>
<math>\Box</math>

=== Proposition 35 ===

If <math>d > 1/\sqrt{2}</math> obeys the inequalities
:<math>\sin( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
and
:<math>\cos( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
then
:<math>p_d \leq \frac{1}{2} - \frac{1}{188}.</math>

(One can check that the conditions are obeyed precisely when <math>d \geq \frac{\sqrt{33}-1}{8} = 0.84307\dots</math>.)

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. Since <math>AB</math> is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the triangle <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>.

Now let <math>BADE</math> be a rhombus with sidelengths d and <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. By the hypotheses, the diagonals BD, AE of this rhombus have length at least 1/2, and hence are monochromatic with probability at most 1/2 by Lemma 2. As above, ABD and BDE are each monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>. As BD is monochromatic with probability at most <math>1/2</math>, we conclude that BADE is monochormatic with probability at least <math>\frac{1}{2}-6\delta</math>.

Now let <math>EDFG</math> be another rhombus congruent to <math>BADE</math>. As BD, AE have length at least 1/2, at least one of the long diagonals BF, AG have length at least 1/2 (the diagonal opposite an obtuse or right-angled triangle will work). Let's say BF has length at least 1/2. As BADE and EDFG are both monochromatic with probability at least <math>\frac{1}{2}-6\delta</math>, and the common edge DE is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the entire configuration ABDEFG is monochromatic with probability at least <math>\frac{1}{2}-11\delta</math>. In particular the pentagon ABDEF is monochromatic with at least this probability. However, in this pentagon, the five edges BA, AD, DE, EB, EF are monochromatic with probability at most <math>\frac{1}{2}-\delta</math>, and the other five edges are monochromatic with probability at most <math>\frac{1}{2}</math> by Lemma 2. Thus the probability that at least one of the edges of this pentagon is monochromatic is at most <math>(\frac{1}{2}-11\delta) + 5 \times 10\delta + 5 \times 11\delta = \frac{1}{2}+94\delta</math>. On the other hand, by the pigeonhole principle, this probability is 1. The claim follows. <math>\Box</math>

=== Proposition 36 ===

If <math>d \ge \frac{2}{\sqrt{15}} = 0.5163\dots</math>, then <math>p_d \leq \frac{1}{2} - \frac{1}{62}</math>,

if <math>d \ge \frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math>, then <math>p_d \leq 0.48</math>,

and <math>\limsup_d p_d\leq \frac{323}{675}=0.4785\ldots</math> (so <math>p_d<0.4786</math> if <math>d</math> is large enough).

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. A simple calculation shows that if <math>d \ge \frac{2}{\sqrt{15}}</math>, then <math>|BD| \ge \frac{1}{2}</math>.
If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. By inclusion-exclusion, we conclude that outside of the event that <math>AB</math> is monochromatic, the probability that <math>AD</math> is monochromatic is at most <math>2\delta</math>.

Now let <math>ADE</math> the <math>180^\circ</math> rotation of <math>ADB</math> around the midpoint of <math>AD</math>, and let <math>FDE</math> be the <math>180^\circ</math> rotation of <math>ADE</math> around the midpoint of <math>DE</math>. By the hypotheses, the line segments <math>AE, BD, BE, BF, DF</math> all have length at least 1/2. Let <math>{\mathcal E}</math> be the event that at least one of <math>AB, AD, DE, EF</math> is monochromatic. By the previous paragraph, this event occurs with probability at most <math>\frac{1}{2}-\delta+2\delta+2\delta+2\delta = \frac{1}{2}+5\delta</math>.

By previous considerations, <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>, and this event lies in <math>{\mathcal E}</math>. On the other hand, <math>BD</math> is monochromatic with probability at most 1/2 by Lemma 2. We conclude that outside of <math>{\mathcal E}</math>, <math>BD</math> is only monochromatic with probability at most <math>3\delta</math>. A similar argument (replacing <math>ABD</math> by <math>DAE</math> or <math>EDF</math>) shows that outside of <math>{\mathcal E}</math>, <math>AE</math> is monochromatic with probability at most <math>3\delta</math>, and similarly for <math>DF</math>. Now we consider <math>EB</math>. By previous considerations, the probability that <math>ABDE</math> is monochromatic is at least <math>\frac{1}{2}-5\delta</math>, and this event lies inside <math>{\mathcal E}</math>. Thus, outside of <math>{\mathcal E}</math>, the probability that <math>EB</math> is monochromatic is at most <math>5\delta</math>; similarly for <math>AF</math>. Finally, the probability that <math>BF</math> is monochromatic outside of <math>{\mathcal E}</math> is at most <math>7\delta</math>. We conclude that outside of an event of probability
:<math>\frac{1}{2}+5\delta + 3\delta+3\delta+3\delta+5\delta+5\delta+7\delta = \frac{1}{2} + 31\delta,</math>
none of the ten edges connecting <math>A,B,D,E,F</math> are monochromatic. But by the pigeonhole principle, this cannot occur in a 4-coloring, hence <math>\frac{1}{2} + 31 \delta \geq 1</math>, and the first claim follows.

For the second claim, we need to use an iterative argument, by feeding the bounds obtained back into the place in the proof where Lemma 2 is currently invoked. To have all occurring distances stay larger than <math>d</math>, we only need to check <math>|BD| \ge d</math>. Equality occurs when <math>ABD</math> is an equilateral triangle, which means that <math>ABC</math> and <math>ACD</math> are isosceles triangles with sides <math>d,d,1</math> and either with angles <math>150^\circ,15^\circ,15^\circ</math>, or with angles <math>30^\circ,75^\circ,75^\circ</math>. From here calculation gives <math>d \ge \frac{1}{2sin(75^\circ)}=\frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math> and <math>d \le \frac{1}{2sin(15^\circ)}=\frac{\sqrt{3}+1}{\sqrt{2}}=1.9318\dots</math>, but the upper bound is not really important, as for us it is enough that <math>|BD|</math> always stay above <math>d_0=\frac{\sqrt{3}-1}{\sqrt{2}}</math>, which occurs everywhere above this value. Now pick a <math>d\ge d_0</math> for which <math>p_d\ge \frac{1}{2}-\delta-\varepsilon</math>, where <math>\sup_{d\ge d_0} p_d= \frac{1}{2}-\delta</math> and <math>\varepsilon</math> is a small positive number. The calculation of the first case gives <math>\frac{1}{2}+5\delta + 2\delta+2\delta+2\delta+4\delta+4\delta+6\delta+O(\varepsilon) =\frac{1}{2} + 25 \delta+O(\varepsilon) \geq 1</math>, from which <math>\delta\ge 0.02</math> if we choose <math>\varepsilon</math> small enough.

To prove the last claim, we modify the construction; we obtain <math>F</math> by reflecting <math>B</math> to <math>AD</math>, to win <math>2\delta</math> in the last step of the calculation. To invoke Lemma 2, we need (among other things) that <math>EF</math> is at least 1/2, and to iterate in a straight-forward way, we would need a value <math>d_0</math> for which <math>EF</math> is at least <math>d_0</math> if <math>AB</math> is <math>d\ge d_0</math>, but such a <math>d_0</math> doesn't exist. We can, however, still iterate in a weaker sense, as <math>6</math> of the occurring <math>10</math> distances tend to infinity as <math>d=|AB|</math> tends to infinity, and the remaining <math>4</math> are also larger than <math>\frac{\sqrt{3}-1}{\sqrt{2}}</math>, so their probability of them being monochromatic is at most <math>0.48=(0.5-\delta)+(\delta-0.02)</math>. What we get eventually is <math>\frac{1}{2} + 25 \delta-2\delta+ 4(\delta-0.02)+O(\varepsilon) =0.42 + 27 \delta+O(\varepsilon) \geq 1</math>, from which <math>p_d\le \frac{323}{675}=0.4785\ldots</math> for <math>d</math> large enough.
<math>\Box</math>

=== Lemma 37 ===

One has <math>\sup_{0 < d < 2} p_d \geq 1/3</math>.

'''Proof''' For a large integer <math>n</math>, consider the points <math>e^{2\pi i j/n}</math> with <math>j=1,\dots,n</math>. Any unit distance coloring will color these points in at most 3 colors, hence divides the n points into three color classes of some size <math>n_1,n_2,n_3</math>. The number of monochromatic pairs is then
:<math>\frac{n_1(n_1-1)}{2} + \frac{n_2(n_2-1)}{2} + \frac{n_3(n_3-1)}{2} = \frac{1}{2} (n_1^2+n_2^2+n_3^2) + O(n) \geq \frac{1}{6} n^2 + O(n)</math>
by Cauchy-Schwarz. Thus at least <math>1/3-O(1/n)</math> of the pairs are monochromatic. Taking expectations and using the pigeonhole principle, we conclude that one of the distances has a probability at least <math>1/3 -O(1/n)</math> of being monochromatic, giving the claim. <math>\Box</math>

=== Lemma 38 ===

Let ABC be a unit-edge equilateral triangle, and let D be an arbitrary point. Let <math>|AD|, |BD|</math> and <math>|CD|</math> be <math>d,e,f</math> respectively. Then <math>p(d)+p(e)+p(f) \leq 1</math>. In particular, examining the case <math>e=f</math>, if <math>p(d) \geq k</math> then <math>p(\sqrt(d(d \pm \sqrt 3) + 1) \leq (1-k)/2</math>.

'''Proof''' At most one of <math>AD, BD</math> and <math>CD</math> can be monochromatic. <math>\Box</math>

Note: A consequence is that a 4-chromatic unit-distance graph G can demonstrate CNP <math>> 4</math> if, for the {d,e,f} arising from some choice of D above, G contains three equal-sized non-empty sets v_d, v_e, v_f of vertex-pairs such that (a) each vertex-pair within v_d is at distance d (resp. e and f), and (b) in any 4-colouring of G, more than 1/3 of the vertex-pairs in the union of the three sets are monochromatic. Note that this demonstration does not require that v_d contain all the vertex-pairs of G that are at distance d (resp. e and f), nor even that the graph {A,B,C,D} which gives rise to {d,e,f} be a subgraph of G. It seems plausible to find such a graph that is small (and/or symmetrical) enough that its colourings can be human-analysed to establish this property.

== Simplification rules for triplets of points in the complex plane ==
Deduced from the rule <math>{\bf P}(A\land B)+{\bf P}(A\land \lnot B)={\bf P}(A)</math>.
:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) = {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) - {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) ) - {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) \neq {\mathbf c}(z_0) ) + {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) = {\mathbf c}(z_0) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

== Proofs of bounds for conditional probabilities ==
The trivial case, valid where <math>\left|d\right|\neq 1</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) = {\mathbf c}(d) )=0</math>

Trivial plus Baye's Theorem, valid where <math>d\neq 0</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) )=\frac{{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )}\leq 1</math>
Rearrange:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )+{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1</math>

Spindle method: for <math>\left|d\right|=d\geq 1/2</math> and <math>\theta=2\text{arcsin}\left(\frac{1}{2d}\right)</math>:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{i\theta}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) ) = \frac{1}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )} - 1\leq 1</math>
which is another way to see <math>\left|d\right|=d\geq 1/2\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1/2</math>.

== Further questions ==
* For <math>n,m\geq CNP</math>, what consistent relationships exist between <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert n\text{ colors}\right)</math> and <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert m\text{ colors}\right)</math>? How can these relationships be used to sharpen arguments of the probabilistic formulation?

[[Category:Polymath16]]

Probabilistic formulation of Hadwiger-Nelson problem

2018-08-01T18:28:56Z

Aubrey: /* Bounds on p_d for 4-colourings */

Suppose for sake of contradiction that we have a 4-coloring <math>c: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with no unit edges monochromatic, thus

:<math>c(z) \neq c(w) \hbox{ whenever } |z-w| = 1. \quad (1)</math>

We can create further such colorings by composing <math>c</math> on the left with a permutation <math>\sigma \in S_4</math> on the left, and with the (inverse of) a Euclidean isometry <math>T \in E(2)</math> on the right, thus creating a new coloring <math>\sigma \circ c \circ T^{-1}: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with the same property. This is an action of the solvable group <math>S_4 \times E(2)</math>.

It is a fact that all solvable groups (viewed as discrete groups) are [https://en.wikipedia.org/wiki/Amenable_group amenable], so in particular <math>S_4 \times E(2)</math> is amenable. This means that there is a finitely additive probability measure <math>\mu</math> on <math>S_4 \times E(2)</math> (with all subsets of this group measurable), which is left-invariant: <math>\mu(gE) = \mu(E)</math> for all <math>g \in S_4 \times E(2)</math> and <math>E \subset S_4 \times E(2)</math>. This gives <math>S_4 \times E(2)</math> the structure of a finitely additive probability space. We can then define a random coloring <math>{\mathbf c}: {\bf C} \to \{1,2,3,4\}</math> by defining <math>{\mathbf c} := {\mathbf \sigma} \circ c \circ {\mathbf T}^{-1}</math>, where <math>({\mathbf \sigma},{\mathbf T})</math> is the element of the sample space <math>S_4 \times E(2)</math>. Thus for any complex number <math>z</math>, the random color <math>{\mathbf c}(z)</math> is a random variable taking values in <math>\{1,2,3,4\}</math>. The left-invariance of the measure implies that for any <math>(\sigma,T) \in S_4 \times E(2)</math>, the coloring <math> \sigma \circ {\mathbf c} \circ T^{-1}</math> has the same law as <math>{\mathbf c}</math>. This gives the color permutation invariance

:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(z_1) = \sigma(c_1), \dots, {\mathbf c}(z_k) = \sigma(c_k) )\quad (2)</math>

for any <math>z_1,\dots,z_k \in {\bf C}</math>, <math>c_1,\dots,c_k \in \{1,2,3,4\}</math>, and <math>\sigma \in S_4</math>, and the Euclidean isometry invariance
:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(T(z_1)) = c_1, \dots, {\mathbf c}(T(z_k)) = c_k. \quad (3)</math>
(In probabilistic language, this means that the random coloring is a [https://en.wikipedia.org/wiki/Stationary_process stationary process] with respect to the action of <math>S_4 \times E(2)</math>. The extraction of a stationary process from a deterministic object is an example of the ''Furstenberg correspondence principle''.)

== Bounds on <math>p_d</math> for 4-colourings ==

A class of correlations that is of particular interest is that of vertex pairs at some distance <math>d</math>. Accordingly, define
:<math>p_d := {\bf P}( \mathbf{c}(0) = \mathbf{c}(d) ).</math>

{| border=1
|-
! distance !! Lower bound !! Lower-bounding graph/method !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math> \geq 1/2</math>
|
|
| 1/2
| Spindle
|
|-
| <math>\geq 2/\sqrt{15}</math>
|
|
| <math>15/31</math>
| Proposition 36
|
|-
| <math>\geq (\sqrt{3}-1)/\sqrt{2}</math>
|
|
| <math>12/25</math>
| Proposition 36
|
|-
| large enough
|
|
| <math>323/675 = 0.4785\ldots</math>
| Proposition 36
|
|-
| <math> d\geq 1/n, n>1</math>
|
|
| <math>1 - 1/n</math>
| (n+1)-gon with one edge length 1 and the rest d, Lemma 34
|
|-
| <math> d\geq 1/(n \sqrt{3}), n>1</math>
|
|
| <math>(3n-2)/3n</math>
| (n+1)-gon with one edge length <math>1/\sqrt{3}</math> and the rest d, Lemma 34
| Not better than the above on intervals <math>\left(\frac{1}{7},\frac{1}{4\sqrt{3}}\right),\left(\frac{1}{4},\frac{1}{2\sqrt{3}}\right)</math>
|-
| <math>1</math>
| <math>0</math>
| Unit edge
| <math>0</math>
| Unit edge
|
|-
| <math>1/\sqrt{3}</math>
| <math>1/28</math>
| Unit diamond plus centres of triangles, together with H, Corollary 16
| <math>1/3</math>
| Unit triangle plus its centre
|
|-
| <math>2</math>
| <math>1/4</math>
| <math>G_{21}</math>
| <math>12/25</math>
| Proposition 36
| Lower bound computer verified
|-
| <math>\sqrt{3}</math>
| <math>13/50</math>
| H, Prop 36
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{7}</math>
|
|
| <math>87/100</math>
| Lemma 38 and Corollary 16
|
|-
| <math>(\sqrt{5}+1)/2</math>
| <math>1/5</math>
| regular pentagon with unit side length
| <math>2/5</math>
| regular pentagon with unit side length
|
|-
| <math>(\sqrt{5}-1)/2</math>
| <math>1/5</math>
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
| <math>2/5</math>
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
|
|-
| <math>\sqrt{11/3}</math>
| <math>1/118</math>
| <math>G_{40}</math>
| <math>12/25</math>
| Proposition 36
| computer-verified
|-
| <math>8/3</math>
| <math>1</math>
| <math>V \oplus V \oplus H</math>
| <math>12/25</math>
| Proposition 36
| computer-verified; leads to contradiction
|-
| <math>(\sqrt{6} \pm \sqrt{2})/2</math>
| <math>1/6</math>
| An arrangement of five vertices
| <math>1/2</math> and | <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{2}</math>
| <math>1/14</math>
| A graph of 13 vertices
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math>
| <math>1/28</math>
| A graph of 13 vertices
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{3/2 + \sqrt{33}/6}</math>
| <math>1/196</math>
| A graph of 9 vertices
| <math>12/25</math>
| Proposition 36
|
|-
| <math>\sqrt{5/3}</math>
| <math>1/756</math>
| A graph of 33 vertices
| <math>12/25</math>
| Proposition 36
|
|-
| <math>2/\sqrt{3}</math>
| <math>1/177</math>
| A graph of 103 vertices
| <math>12/25</math>
| Proposition 36
| Computer-verified
|-
| <math>(\sqrt{33} \pm 1)/(2\sqrt{3})</math>
| <math>1/2</math>
| <math>G_{420}</math>
| <math>12/25</math>
| Proposition 36
| Computer-verified
|}

== Bounds on <math>{\bf P}( \mathbf{c}(0) = \mathbf{c}(d_1) \mid \mathbf{c}(0) \neq \mathbf{c}(d_0) )</math> for 4-colourings ==

{| border=1
|-
! <math>d_1</math> !! <math>d_0</math> !! Lower bound !! Lower-bounding graph !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math>1+(-1)^{1/3}</math>
| <math>2</math>
| 1/2
| H
|
|
| Equals <math>p_{\sqrt 3}/(1-p_2)</math>
|-
| <math>1+(-1)^{1/3}</math>
| <math>1+(-1)^{-1/3}</math>
|
|
| 1/2
| H
|
|}

== Proofs of bounds ==

One can compute some correlations of the coloring exactly:

=== Lemma 1 ===
Let <math>z,w \in {\bf C}</math> with <math>|z-w|=1</math>. Then
:<math> {\bf P}( \mathbf{c}(z) = c ) = \frac{1}{4}\quad (4)</math>
for all <math>c=1,\dots,4</math>,
:<math> {\bf P}( \mathbf{c}(z) = \mathbf{c}(w) ) = 0\quad (5),</math>
and
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c' ) = \frac{1}{12} \quad (6)</math>
for any distinct <math>c,c' \in \{1,2,3,4\}</math>. If <math>u</math> is at a unit distance from both z and w, then
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c'; \mathbf{c}(u) = c'' ) = \frac{1}{24} \quad (6')</math>

'''Proof''' By color invariance (2), the four probabilities in (4) are equal and sum to 1, giving (4). The claim (5) is immediate from (1). From (5) and color invariance, the 12 probabilities in (6) are equal and sum to 1, giving (6). The same argument gives (6').<math>\Box</math>

=== Lemma 2 ===
(Spindle argument) Let <math>|d| \geq 1/2</math>. Then
:<math> p_d \leq \frac{1}{2} \quad (7).</math>

'''Proof''' We can find an angle <math>\theta</math> with <math>|de^{i\theta}-d|=1</math>, then <math>\mathbf{c}(de^{i\theta}) \neq \mathbf{c}(d)</math> almost surely. This means that at least one of the events <math>\mathbf{c}(0) = \mathbf{c}(d)</math>, <math>\mathbf{c}(0) = \mathbf{c}(d e^{i\theta})</math> occurs with probability at most 1/2. The claim now follows from isometry invariance (3). <math>\Box</math>

=== Lemma 3 ===
(Using the K graph) We have
:<math>52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) + {\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} ) \geq 1 \quad (8).</math>

'''Proof''' Consider the 61-vertex graph <math>K</math> from [https://arxiv.org/abs/1804.02385 de Grey's paper]. It has 26 (isometric) copies of H, and thus 52 copies of the triangle <math>(1, e^{2\pi i/3}, e^{4\pi i/3})</math>. With probability at least <math>1 - 52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) </math>, none of these triangles are monochromatic. By the argument in that paper, this implies that the three linking diagonals <math>(-2, +2)</math>, <math>(-2 e^{2\pi i/3}, 2e^{2\pi i/3})</math>, <math>(-2 e^{4\pi i/3}, e^{-4\pi i/3})</math> are monochromatic. This gives the claim. <math>\Box</math>

=== Corollary 4 ===
(Existence of monochromatic <math>\sqrt{3}</math>-triangles) We have <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) \geq \frac{1}{104}</math>.

'''Proof''' The probability <math>{\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} )</math> is at most <math>{\bf P}( \mathbf{c}(-2) = \mathbf{c}(2)) = p_4</math>, which by Lemma 2 is at most 1/2. The claim now follows from Lemma 3. <math>\Box</math>

=== Computer-verified Claim 5 ===
(Using the graph M) One has <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = 0</math> (Note this contradicts Corollary 4).

'''Proof''' This simply reflects the fact that there is no 4-coloring of the 1345-vertex graph M from [https://arxiv.org/abs/1804.02385 de Grey's paper] with its central copy of H containing a monochromatic triangle. One can use other graphs for this purpose, such as the 278-vertex graph <math>M_1</math> or <math>V \oplus V \oplus H</math>. <math>\Box</math>

=== Computer-verified Claim 6 ===
(Using the graph <math>V \oplus V \oplus H</math>) One has <math>p_{8/3} = 1</math> (note this contradicts Lemma 2).

'''Proof''' This reflects the fact that every 4-coloring of <math>V \oplus V \oplus H</math> must assign the same color to 0 and 8/3. There is also a 745-vertex subgraph of <math>V \oplus V \oplus H</math> with the same property. <math>\Box</math>

=== Lemma 7 ===
(Using <math>G_{40}</math>) We have
:<math>59 p_{\sqrt{11/3}} + p_{8/3} \geq 1</math>.

'''Proof''' This reflects the fact that every 4-coloring of the 40-vertex graph <math>G_{40}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which none of the 59 pairs of vertices at distance <math>\sqrt{11/3}</math> apart, will assign the same color to 0 and 8/3. (This is presumably human-verifiable.) <math>\Box</math>

=== Corollary 8 ===
One has
:<math> p_{\sqrt{11/3}} \geq \frac{1}{118}</math>.

'''Proof''' Combine Lemma 7 and Lemma 2. <math>\Box</math>

=== Lemma 9 ===
(Using <math>G_{49}</math>) One has
:<math>18 {\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq p_{\sqrt{11/3}} .</math>

'''Proof''' This reflects the fact that every 4-coloring of the 49-vertex graph <math>G_{49}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which 0 and <math>\sqrt{11/3}</math> have the same color, at least one of the 18 copies of <math>(1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3)</math> is monochromatic. This is potentially human-verifiable. <math>\Box</math>

=== Corollary 10 ===
(Existence of monochromatic <math>1/\sqrt{3}</math>-triangles) One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq \frac{1}{2124}.</math>

'''Proof''' Combine Corollary 8 and Lemma 9. <math>\Box</math>

=== Computer-verified claim 11 ===
One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) = 0</math>. (This contradicts Corollary 10).

'''Proof''' This reflects the fact that the 627-vertex graph <math>G_{627}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] does not have any 4-colorings with <math>1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3</math> monochromatic. <math>\Box</math>

=== Lemma 12 ===
For certain special distances d, one can improve the bound in Lemma 2:

If <math>k \geq 1</math> is a natural number, <math>j\in\mathbb{Z}</math>, <math>\gcd(j,2k+1)=1</math>, and <math>r = \frac{1}{2} \csc\left(\frac{j\pi}{2k+1}\right)</math> then
:<math> p_r \leq \frac{k}{2k+1},</math>
thus for instance
:<math> p_{\frac{1}{\sqrt{3}}} \leq \frac{1}{3}.</math>

'''Proof''' Observe that the regular 2k+1-polygon <math>r, re^{2\pi i/(2k+1)}, r e^{4\pi i/(2k+1)}, \dots, r^{4k\pi i/(k+1)}</math> has unit side lengths. By the pigeonhole principle, we conclude that at most k of these vertices can have the same color as the origin, and the claim follows. <math>\Box</math>

On the other hand, for <math>k=2,j=1</math> we also know from the regular pentagon of unit sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}+1}{2}} \leq \frac{2}{5} \quad (9)</math>
since any 4-coloring of that pentagon has either one or two monochromatic diagonals.

Similarly, for <math>k=2,j=2</math> we also know from the regular pentagon of <math>\frac{\sqrt{5}-1}{2}</math> sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}-1}{2}} \leq \frac{2}{5}</math>
since any 4-coloring of that pentagon has either one or two monochromatic edges. More generally, if <math>a,b,c,d,e</math> are the diagonal lengths of a pentagon with unit sides, then
:<math> 1 \leq p_a + p_b + p_c + p_d + p_e \leq 2.</math>

=== Lemma 13 ===
We have
:<math> 7 p_{\frac{1}{\sqrt{3}}} \geq p_{\sqrt{3}}</math>.

'''Proof''' Consider the unit rhombus <math>0, 1, e^{i\pi/3}, e^{-i\pi/3}</math> together with the centers <math>e^{i\pi/6}/\sqrt{3}, e^{-i\pi/6}/\sqrt{3}</math>. With probability <math>p_{\sqrt{3}}</math>, the two far vertices <math>e^{i\pi/3}, e^{-i\pi/3}</math> are the same color, and then 0,1 will be two other colors. This forces either one of the centers <math>e^{i\pi/6}/\sqrt{3}</math> of a triangle to have a common color with one of the vertices of that triangle, or the two centers must have the same color. Thus in any event one of the seven edges of distance <math>1/\sqrt{3}</math> is monochromatic, giving the claim. <math>\Box</math>

=== Corollary 14 ===
We have <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{728}</math>.

This slightly improves upon the lower bound of 1/2124 coming from Corollary 10.

'''Proof''' Combine Corollary 4 and Lemma 13. <math>\Box</math>

=== Lemma 15 ===
One has
:<math>p_{\sqrt{3}} + p_2 \geq \frac{2}{3}</math> and <math>2 p_{\sqrt{3}} + p_2 \geq 1</math>.

'''Proof''' As noted in de Grey's paper, there are essentially four 4-colorings of H. H has six edges of length <math>\sqrt{3}</math> and three of length <math>2</math>. If we let a denote the number of monochromatic <math>\sqrt{3}</math> edges and b the number of monochromatic <math>2</math>-edges, we see from inspection of all four colorings that <math>(a,b)</math> is either <math>(6, 0), (4,0), (2, 1)</math>, or <math>(0,3)</math>. In particular, one always has <math>\frac{a}{6} + \frac{b}{3} \geq \frac{2}{3}</math> and <math>2\frac{a}{6} + \frac{b}{3} \geq 1</math>. Taking expectations, we obtain the claim. <math>\Box</math>

=== Corollary 16 ===
We have <math>p_2 \geq \frac{1}{6}</math>, <math>p_{\sqrt{3}} \geq \frac{1}{4} </math> and <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{28}</math>.

'''Proof''' Combine Lemma 2, Lemma 15, and Lemma 13. <math>\Box</math>

=== Lemma 17 ===
If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths a,b,c. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also, thus
:<math>{\bf P}(\mathbf{c}(0) \neq \mathbf{c}(a)) + {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(b)) \geq {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(c))</math>
and the claim follows. <math>\Box</math>

Note that Lemma 2 follows from the a=b, c=1 case of this lemma. Iterating this lemma starting with Lemma 2 we can also obtain slightly nontrivial upper bounds on <math>p_a</math> for small values of a, e.g. <math>p_a \leq 1 - 2^{-k}</math> when <math>a \geq 2^{-k}, k\in\mathbb{Z}^+</math>.

Further, we can generalise the a=b case to one in which the triangle is replaced by a (k+1)-gon of which one edge is 1 and the others are all equal, leading to the stronger result <math>p_a \leq 1 - 1/k</math> when <math>a \geq 1/k, k\in\mathbb{Z}^+ \land k>1</math>. Further strengthening is achieved by using <math>1/\sqrt{3}</math> as the long edge, given Lemma 12.

=== Lemma 18 ===
Whenever <math>d>0</math>, one has the inequalities
:<math> |p_{\phi d} - p_d| \leq \frac{2}{5}, p_{\phi d} + p_d \geq \frac{1}{5}, 2p_d - p_{\phi d} \leq 1, 2 p_{\phi d} - p_d \leq 1</math>
where <math>\phi := \frac{1+\sqrt{5}}{2}</math> is the golden ratio. Also we have
:<math> p_{d/\sqrt{3}} \leq \frac{1}{3} + p_d, \frac{1}{2} + \frac{1}{2} p_d.</math>

Note that this generalises (9), as well as a special case of Lemma 12.

'''Proof''' Consider the regular pentagon with sidelength <math>d</math>, so it also has 5 diagonals of length <math>\phi d</math>. Let <math>a \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic edges and let <math>b \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic diagonals. Observe:
* <math>a,b</math> cannot both be zero (pigeonhole principle).
* <math>a</math> cannot be 4. Similarly, <math>b</math> cannot be 4.
* If <math>a=5</math>, then <math>b=5</math>, and conversely.
* If <math>a=0</math>, then <math>b=1,2</math>; similarly, if <math>b=0</math>, then <math>a=1,2</math>.

From this we observe the inequalities
:<math> |\frac{a}{5}-\frac{b}{5}| \leq \frac{2}{5}; \frac{a}{5} + \frac{b}{5} \geq \frac{1}{5}; 2 \frac{a}{5} - \frac{b}{5} \leq 1; 2\frac{b}{5} - \frac{a}{5} \leq 1</math>
and on taking expectations we obtain the first claim. Similarly, if one considers the colorings of an equilateral triangle of sidelength <math>d</math> together with its center, and counts the numbers <math>a,b \in \{0,1,2,3\}</math> of monochromatic edges of length <math>d</math> and <math>d/\sqrt{3}</math> respectively, one observes that one always has <math>\frac{b}{3} \leq \frac{1}{3} + \frac{2}{3} \frac{a}{3}, \frac{1}{2} + \frac{1}{2} \frac{a}{3}</math>, and on taking expectations one obtains the claim.<math>\Box</math>

The hexagon <math>H</math> has essentially four distinct colorings: the coloring <math>\hbox{2tri}</math> with two triangles, the coloring <math>\hbox{1tri}</math> with one triangle, the coloring <math>\hbox{axisym}</math> that is symmetric around an axis, and the coloring <math>\hbox{centralsym}</math> that is symmetric around the central point. This gives four probabilities <math>p_{H = 2tri}, p_{H = 1tri}, p_{H = axisym}, p_{H = centralsym}</math> that sum to 1. By counting the number of monochromatic edges of length <math>\sqrt{3}, 2</math> respectively, one also obtains the identities
:<math>p_{\sqrt{3}} = p_{H = 2tri} + \frac{2}{3} p_{H = 1tri} + \frac{1}{3} p_{H = axisym}; \quad p_2 = \frac{1}{3} p_{H=axisym} + p_{H=centralsym}</math>
which reproves Lemma 15. Also
:<math> {\bf P}( \mathbf{c}(0) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = p_{H = 2tri} + \frac{1}{2} p_{H=1tri}.</math>
Any 4-coloring of L contains at least one triangle within one of its 52 copies of H, thus
:<math> p_{H = 2tri} + \frac{1}{2} p_{H=1tri} \geq \frac{1}{52}</math>
which reproves Corollary 4.

=== Lemma 19 ===
(Hubai) One has <math>p_{H = 1tri} + p_{H = axisym} \geq \frac{1}{10}</math>.

'''Proof''' Consider five copies of H centred at 0,1,2,3,4. With probability at least <math>1 - 5( p_{H = 1tri} + p_{H = axisym} )</math>, none of these copies of H are colored 1tri or axisym, and so must be colored 2tri or centralsym. One can check then that if one of the copies is colored 2tri, then so is any adjacent copy; thus all five copies are colored 2tri, or all five are colored centralsym. In either case we see that -1 and 5 are colored the same color. Comparing with Lemma 2 then gives the claim. <math>\Box</math>

=== Theorem 20 ===
One has <math>p_{H = 1tri} > 0</math>.

'''Proof''' Suppose for contradiction that <math>p_{H = 1tri} = 0</math>. One can then run a version of the de Bruijn-Erdos argument to obtain a coloring in which 1tri hexagons are completely nonexistent (since there are arbitrarily large finite colorings with this property). Consider the triangular lattice <math>{\bf Z}[e^{\pi i/3}]</math>. We 2-color the edges of this lattice by coloring an edge black if it is the short diagonal of a unit rhombus with monochromatic long diagonal, and white otherwise. The four colorings of hexagons lead to four possible colorings at each vertex:

* If H is colored 2tri, then all six edges to the centre of H are black.
* If H is colored 1tri, then two edges to the centre of H at 120 degree angles are white, the other four are black.
* If H is colored axisym, then two opposing edges of the centre of H are black, the other four are white.
* If H is colored centralsym, then all six edges to the centre of H are black.

In particular, as we are assuming no 1tri hexagons, the faces cut out by the black edges have angles 60 degrees, and thus must be equilateral triangles, sectors of angle 60, half-planes, or the entire plane. If there is at least one equilateral triangle, then the rest of the black edges must form an equilateral lattice with that triangle sidelength. This leads to only a small number of possible hexagon colorings in the lattice:

# Case 1: All edges white.
# Case 2: All edges black.
# Case 3.k: For some natural number <math>k \geq 2</math>, the length k edges joining adjacent vertices in some coset of <math>k \cdot {\mathbf Z}[ e^{\pi i/3} ]</math> are all black, and the remaining edges are white.
# Case 4: Each horizontal row consists either entirely of black edges, or entirely of white edges.
# Case 5: Each northwest row consists either entirely of black edges, or entirely of white edges.
# Case 6: Each northeast row consists either entirely of black edges, or entirely of white edges.
# Case 7: Six rays of black edges meeting at a common vertex; all other edges white.

Technically, Case 1 is contained in Cases 4,5,6 as written above, but this will not be an issue. One can view Case 7 as a limiting case <math>k=\infty</math> of Case 3.k; Case 2 is similarly the opposite limiting case <math>k=1</math>.

In the first case, the coloring is periodic with periods <math>2, 2 e^{\pi i/3}</math>. In the second case, it is periodic with periods <math>3, 3 e^{\pi i/3}</math>. In the third case, it is periodic with periods <math>3k, 3k e^{\pi i/3}</math>. Also note that for each k, one can check if Case 3.k holds by inspecting the coloring at a finite number of vertices. Thus the event that Case 3.k holds is "measurable" in the sense that a meaningful probability can be assigned. (But Cases 1,2,4,5,6 are not measurable events, they require an infinite number of points to be inspected, and the probability measure we are using is only finitely additive rather than infinitely additive.) In Case 4, the coloring is periodic with period 2; also, every coset of <math>2 \cdot {\bf Z}[e^{\pi i/3}]</math> is 2-colored. Similarly for Case 5 and 6 (where the periods are <math>2 e^{2\pi i/3}</math> and <math>2 e^{4\pi i/3}</math> respectively.)

Let <math>\alpha_k</math> be the probability that Case 3.k holds for the given value of <math>k</math>. Then <math> \sum_{k=2}^K \alpha_k \leq 1</math> for any k, hence <math>\sum_{k=2}^\infty \alpha_k \leq 1</math>. In particular, we can find <math>K_1</math> such that <math>\sum_{k={K_1}}^\infty \alpha_k \leq 0.1</math> (say). Let <math>P</math> be six times the least common multiple of <math>1,2,\dots,K_1</math>. Then the coloring is P- and <math>P e^{\pi i/3}</math>-periodic for Case 1, Case 2, and all Case 3.k with <math>k \leq K_1</math>. On the other hand, if <math>K_2</math> is sufficiently large depending on <math>P</math>, and Case 3.k holds for some <math>k \geq K_2</math>, then almost all of the hexagons are colored centralsym, which makes the coloring "almost <math>P, P e^{\pi i/3}</math>-periodic" in the sense that
:<math>{\bf c}(z+P e^{\pi i j/3}) = {\bf c}(z) \hbox{ for } j=0,1,2,3,4,5</math>
will hold for at least <math>0.9</math> of the lattice points <math>z \in {\bf Z}[e^{\pi i/3}]</math> with <math>|z| \leq K_2</math>. Similarly for Case 7 (which is sort of a <math>k=\infty</math> limiting case of Case 3.k.) Thus, with the probability <math> \geq 1 - \sum_{k=K_1}^{K_2} \alpha_k \geq 0.9</math>, the coloring of the seven vertices <math>{\bf c}(0), {\bf c}(P e^{\pi ij/3}, j=1,\dots,6</math> is (up to rotation and recoloring) one of the three patterns of the central and linking vertices in Figure 3 of Aubrey's paper, namely

* <math>{\bf c}(0) = {\bf c}(P) = {\bf c}(P e^{\pi i/3}) = {\bf c}(P e^{2\pi i/3}) = {\bf c}(P e^{3\pi i/3}) = {\bf c}(P e^{4\pi i/3}) = {\bf c}(P e^{5\pi i/3}) </math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{3\pi i/3})</math>; <math>{\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{3\pi i/3})</math>; <math> {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>.

Using the spindling argument from Aubrey's paper, we conclude that the third possibility must in fact hold with probability at least 0.8; on the other hand, from Lemma 2 this scenario can only occur with probability at most 1/2, giving the required contradiction.
<math>\Box</math>

One should be able to refine this argument to show that <math>p_{H = 1tri} > c</math> for an absolute constant <math> c>0 </math>.

=== Lemma 21 ===
Providing a tighter bound for Lemma 17 with a more thorough proof: If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths <math>a,b,c</math>. Let <math>\left|z_2\right|=b,\left|a-z_2\right|=c</math>. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also: <math>\mathbf{c}(a)\neq\mathbf{c}(z_2)\Rightarrow[\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)]</math>. <math>[A\Rightarrow B]\Rightarrow {\bf P}(A)\leq{\bf P}(B)</math> thus
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) \geq {\bf P}(\mathbf{c}(a) \neq \mathbf{c}(z_2)) = 1-p_c</math>
<math>{\bf P}(A\lor B) +{\bf P}(A\land B)={\bf P}(A)+{\bf P}(B)</math>, so
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)) + {\bf P}(\mathbf{c}(0)\neq\mathbf{c}(z_2)) - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>1-p_c \leq 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
Using the law of cosines: <math>z_2=b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)</math>
:<math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math> <math>\Box</math>

=== Lemma 22 ===

One has <math>3 p_{1/\sqrt{3}} \geq {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) )</math>.

'''Proof''' Let <math>z</math> be a complex number of magnitude <math>1/\sqrt{3}</math> that is a unit distance from 1. If <math>\mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) = c</math> (say), then <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> cannot be colored with <math>c</math>; also, <math>z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> are the vertices of a unit equilateral triangle and thus must take on three different colors. By the pigeonhole principle, one of <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> must then take the same color as the origin, and the claim follows. <math>\Box</math>

=== Lemma 23 ===

One has <math>4 p_{(\sqrt{6} \pm \sqrt{2})/2} + p_{\sqrt{2}} \geq 1</math>. In particular, by Lemma 2, <math>p_{(\sqrt{6}+\sqrt{2})/2} \geq 1/8</math>.

'''Proof''' [ExIs2018b] We just prove the claim for the + sign (the - sign can then be obtained after applying the Galois conjugacy that maps <math>\sqrt{3}</math> to <math>-\sqrt{3}</math>, leaving <math>\sqrt{2}</math> unchanged). Set <math>d := \frac{\sqrt{6}+\sqrt{2}}{2}</math>, and consider the five vertices

:<math>0, e^{5\pi i/4}, e^{5\pi i/4} + d, e^{5\pi i/4} + e^{\pi i/3} d, e^{5\pi i/4} + (e^{\pi i/3}-i)d.</math>

One can check that of the ten edges determined by these five vertices, five have unit length, four have length d, and the remaining distance (from 0 to <math>e^{5\pi i/4}+d</math>) has distance <math>\sqrt{2}</math>. Since <math>\mathbf{c}</math> must make one of the latter five edges monochromatic, the claim follows. <math>\Box</math>

=== Lemma 24 ===

One has <math>p_{\sqrt{2}} \geq \frac{1}{14}</math>.

'''Proof''' In page 7 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 20 unit distance edges and 14 edges of length <math>\sqrt{2}</math> is constructed. The coloring <math>\mathbf{c}</math> must make one of the 14 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 25 ===

Let <math>d = \frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math> and <math>e = \frac{3^{1/4} \sqrt{2} + \sqrt{3} - 1}{2}</math>.
Then one has <math>14 p_d + p_e \geq 1</math>. In particular, by Lemma 2, <math>p_d \geq 1/28</math>.

'''Proof''' In page 9 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 19 unit edges, 14 edges of length d, and one edge of length e is constructed. The coloring <math>\mathbf{c}</math> must make one of the 15 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 26 ===

Let <math>d = \sqrt{3/2 + \sqrt{33}/6}</math>. Then <math>7 p_d \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_d \geq \frac{1}{196}</math>.

'''Proof''' In page 11 of [ExIs2018b], a graph of nine vertices consisting of 12 unit edges and 7 edges of length d is constructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Thus, <math>\mathbf{c}</math> can only make the AB edge monochromatic if one of the seven length d edges is monochromatic. The claim follows.
<math>\Box</math>

=== Lemma 27 ===

One has <math>27 p_{\sqrt{5/3}} \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_{\sqrt{5/3}} \geq \frac{1}{756}</math>.

'''Proof''' In page 13 of [ExIs2018], a graph of 33 vertices with some unit edges and 27 edges of length <math>\sqrt{5/3}</math> is contructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Now repeat the proof of Lemma 26. <math>\Box</math>

=== Computer-verified claim 28 ===

One has <math>p_{2/\sqrt{3}} \geq \frac{1}{177}</math>.

'''Proof''' In page 15 of [ExIs2018], a 5-chromatic graph of 103 vertices, 312 unit edges, and 177 edges of length <math>2/\sqrt{3}</math> is constructed. <math>\mathbf{c}</math> must make one of the latter edges monochromatic, giving the claim. <math>\Box</math>

=== Lemma 29 ===

One has <math>p_{(\sqrt{6} \pm \sqrt{2})/2} \geq 1/6</math> (this improves the bound in Lemma 23).

'''Proof''' Use graphs 505 and 507 from [S2004] and the spindle bound. <math>\Box</math>

=== Lemma 30 ===
For <math>m > n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' <math>n</math> colors and <math>m</math> points necessitates at least 2 having equal color. I.e.

:<math>{\bf P}\left(\bigvee_{k=0}^n \bigvee_{j=k+1}^n\ \mathbf{c}(z_k) = \mathbf{c}(z_j)\right) = 1</math>

The lemma then follows immediately from the fact:
:<math>{\bf P}\left(\bigcup_{k} E_k\right) \leq \sum_{k} {\bf P}\left(E_k\right) \,\Box</math>

=== Corollary 31 ===
Let <math>\lvert z_k\rvert=1</math>. Then for <math>m \geq n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' Use lemma 30 on the set <math>\left\{z_k \bigg\vert 1\leq k\leq m \land k\in\mathbb{Z}\right\}\cup\{0\}</math>. Simplify using <math>{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(0) \right)=0</math>.
<math>\Box</math>

=== Lemma 32 ===
For <math>x\in\mathbb{R}</math> and an <math>n</math>-coloring of the plane, <math>\sum_{k=1}^{n-1}\left(n-k\right){\bf P}\left(\mathbf{c}\left(0\right) = \mathbf{c}\left( 2\sin\left(\frac{kx}{2}\right) \right) \right) \geq 1</math>.

'''Proof''' Use corollary 31 on the set <math>\left\{e^{ikx} \bigg\vert 0\leq k < n \land k\in\mathbb{Z}\right\}</math>. and simplify by grouping lengths.
<math>\Box</math>

=== Corollary 33 ===
Interesting(easy to simplify results of) values for <math>x</math> in Lemma 32 are in <math>\left\{x \bigg\vert \sin\left(\frac{kx}{2}\right)=1 \land 1\leq k < n \land k\in\mathbb{Z}\right\}</math>.
For 4-colorings, this gives
:<math>2p_{\sqrt 3}+p_2 \geq 1</math>
:<math>3p_{(\sqrt 3-1)/\sqrt 2}+p_{\sqrt 2} \geq 1</math>
:<math>3p_{2\sin(\pi/18)}+2p_{2\sin(\pi/9)} \geq 1</math>

=== Lemma 34 ===
Generalizing the note of Lemma 17, <math>\lvert d_1\rvert= d_1 > \lvert d_0\rvert= d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>

'''Proof''' let <math>\lvert z_{j+1} -z_j\rvert=d_0 > 0, \lvert z_{j+n} -z_0\rvert=d_1>0</math>.

Base case, <math>n=2</math>, by Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle:
:<math>2d_0\geq d_1\Rightarrow 2p_{d_0}\leq 1+p_{d_1}</math>
The inductive step is Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle. After induction:
:<math>[n\geq 2\land nd_0\geq d_1]\Rightarrow np_{d_0}\leq n-1+p_{d_1}</math>
Substitute <math>n=\left\lceil\frac{d_1}{d_0}\right\rceil</math>, simplify, rearrange:
:<math>d_1 > d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>
<math>\Box</math>

=== Proposition 35 ===

If <math>d > 1/\sqrt{2}</math> obeys the inequalities
:<math>\sin( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
and
:<math>\cos( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
then
:<math>p_d \leq \frac{1}{2} - \frac{1}{188}.</math>

(One can check that the conditions are obeyed precisely when <math>d \geq \frac{\sqrt{33}-1}{8} = 0.84307\dots</math>.)

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. Since <math>AB</math> is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the triangle <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>.

Now let <math>BADE</math> be a rhombus with sidelengths d and <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. By the hypotheses, the diagonals BD, AE of this rhombus have length at least 1/2, and hence are monochromatic with probability at most 1/2 by Lemma 2. As above, ABD and BDE are each monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>. As BD is monochromatic with probability at most <math>1/2</math>, we conclude that BADE is monochormatic with probability at least <math>\frac{1}{2}-6\delta</math>.

Now let <math>EDFG</math> be another rhombus congruent to <math>BADE</math>. As BD, AE have length at least 1/2, at least one of the long diagonals BF, AG have length at least 1/2 (the diagonal opposite an obtuse or right-angled triangle will work). Let's say BF has length at least 1/2. As BADE and EDFG are both monochromatic with probability at least <math>\frac{1}{2}-6\delta</math>, and the common edge DE is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the entire configuration ABDEFG is monochromatic with probability at least <math>\frac{1}{2}-11\delta</math>. In particular the pentagon ABDEF is monochromatic with at least this probability. However, in this pentagon, the five edges BA, AD, DE, EB, EF are monochromatic with probability at most <math>\frac{1}{2}-\delta</math>, and the other five edges are monochromatic with probability at most <math>\frac{1}{2}</math> by Lemma 2. Thus the probability that at least one of the edges of this pentagon is monochromatic is at most <math>(\frac{1}{2}-11\delta) + 5 \times 10\delta + 5 \times 11\delta = \frac{1}{2}+94\delta</math>. On the other hand, by the pigeonhole principle, this probability is 1. The claim follows. <math>\Box</math>

=== Proposition 36 ===

If <math>d \ge \frac{2}{\sqrt{15}} = 0.5163\dots</math>, then <math>p_d \leq \frac{1}{2} - \frac{1}{62}</math>,

if <math>d \ge \frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math>, then <math>p_d \leq 0.48</math>,

and <math>\limsup_d p_d\leq \frac{323}{675}=0.4785\ldots</math> (so <math>p_d<0.4786</math> if <math>d</math> is large enough).

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. A simple calculation shows that if <math>d \ge \frac{2}{\sqrt{15}}</math>, then <math>|BD| \ge \frac{1}{2}</math>.
If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. By inclusion-exclusion, we conclude that outside of the event that <math>AB</math> is monochromatic, the probability that <math>AD</math> is monochromatic is at most <math>2\delta</math>.

Now let <math>ADE</math> the <math>180^\circ</math> rotation of <math>ADB</math> around the midpoint of <math>AD</math>, and let <math>FDE</math> be the <math>180^\circ</math> rotation of <math>ADE</math> around the midpoint of <math>DE</math>. By the hypotheses, the line segments <math>AE, BD, BE, BF, DF</math> all have length at least 1/2. Let <math>{\mathcal E}</math> be the event that at least one of <math>AB, AD, DE, EF</math> is monochromatic. By the previous paragraph, this event occurs with probability at most <math>\frac{1}{2}-\delta+2\delta+2\delta+2\delta = \frac{1}{2}+5\delta</math>.

By previous considerations, <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>, and this event lies in <math>{\mathcal E}</math>. On the other hand, <math>BD</math> is monochromatic with probability at most 1/2 by Lemma 2. We conclude that outside of <math>{\mathcal E}</math>, <math>BD</math> is only monochromatic with probability at most <math>3\delta</math>. A similar argument (replacing <math>ABD</math> by <math>DAE</math> or <math>EDF</math>) shows that outside of <math>{\mathcal E}</math>, <math>AE</math> is monochromatic with probability at most <math>3\delta</math>, and similarly for <math>DF</math>. Now we consider <math>EB</math>. By previous considerations, the probability that <math>ABDE</math> is monochromatic is at least <math>\frac{1}{2}-5\delta</math>, and this event lies inside <math>{\mathcal E}</math>. Thus, outside of <math>{\mathcal E}</math>, the probability that <math>EB</math> is monochromatic is at most <math>5\delta</math>; similarly for <math>AF</math>. Finally, the probability that <math>BF</math> is monochromatic outside of <math>{\mathcal E}</math> is at most <math>7\delta</math>. We conclude that outside of an event of probability
:<math>\frac{1}{2}+5\delta + 3\delta+3\delta+3\delta+5\delta+5\delta+7\delta = \frac{1}{2} + 31\delta,</math>
none of the ten edges connecting <math>A,B,D,E,F</math> are monochromatic. But by the pigeonhole principle, this cannot occur in a 4-coloring, hence <math>\frac{1}{2} + 31 \delta \geq 1</math>, and the first claim follows.

For the second claim, we need to use an iterative argument, by feeding the bounds obtained back into the place in the proof where Lemma 2 is currently invoked. To have all occurring distances stay larger than <math>d</math>, we only need to check <math>|BD| \ge d</math>. Equality occurs when <math>ABD</math> is an equilateral triangle, which means that <math>ABC</math> and <math>ACD</math> are isosceles triangles with sides <math>d,d,1</math> and either with angles <math>150^\circ,15^\circ,15^\circ</math>, or with angles <math>30^\circ,75^\circ,75^\circ</math>. From here calculation gives <math>d \ge \frac{1}{2sin(75^\circ)}=\frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math> and <math>d \le \frac{1}{2sin(15^\circ)}=\frac{\sqrt{3}+1}{\sqrt{2}}=1.9318\dots</math>, but the upper bound is not really important, as for us it is enough that <math>|BD|</math> always stay above <math>d_0=\frac{\sqrt{3}-1}{\sqrt{2}}</math>, which occurs everywhere above this value. Now pick a <math>d\ge d_0</math> for which <math>p_d\ge \frac{1}{2}-\delta-\varepsilon</math>, where <math>\sup_{d\ge d_0} p_d= \frac{1}{2}-\delta</math> and <math>\varepsilon</math> is a small positive number. The calculation of the first case gives <math>\frac{1}{2}+5\delta + 2\delta+2\delta+2\delta+4\delta+4\delta+6\delta+O(\varepsilon) =\frac{1}{2} + 25 \delta+O(\varepsilon) \geq 1</math>, from which <math>\delta\ge 0.02</math> if we choose <math>\varepsilon</math> small enough.

To prove the last claim, we modify the construction; we obtain <math>F</math> by reflecting <math>B</math> to <math>AD</math>, to win <math>2\delta</math> in the last step of the calculation. To invoke Lemma 2, we need (among other things) that <math>EF</math> is at least 1/2, and to iterate in a straight-forward way, we would need a value <math>d_0</math> for which <math>EF</math> is at least <math>d_0</math> if <math>AB</math> is <math>d\ge d_0</math>, but such a <math>d_0</math> doesn't exist. We can, however, still iterate in a weaker sense, as <math>6</math> of the occurring <math>10</math> distances tend to infinity as <math>d=|AB|</math> tends to infinity, and the remaining <math>4</math> are also larger than <math>\frac{\sqrt{3}-1}{\sqrt{2}}</math>, so their probability of them being monochromatic is at most <math>0.48=(0.5-\delta)+(\delta-0.02)</math>. What we get eventually is <math>\frac{1}{2} + 25 \delta-2\delta+ 4(\delta-0.02)+O(\varepsilon) =0.42 + 27 \delta+O(\varepsilon) \geq 1</math>, from which <math>p_d\le \frac{323}{675}=0.4785\ldots</math> for <math>d</math> large enough.
<math>\Box</math>

=== Lemma 37 ===

One has <math>\sup_{0 < d < 2} p_d \geq 1/3</math>.

'''Proof''' For a large integer <math>n</math>, consider the points <math>e^{2\pi i j/n}</math> with <math>j=1,\dots,n</math>. Any unit distance coloring will color these points in at most 3 colors, hence divides the n points into three color classes of some size <math>n_1,n_2,n_3</math>. The number of monochromatic pairs is then
:<math>\frac{n_1(n_1-1)}{2} + \frac{n_2(n_2-1)}{2} + \frac{n_3(n_3-1)}{2} = \frac{1}{2} (n_1^2+n_2^2+n_3^2) + O(n) \geq \frac{1}{6} n^2 + O(n)</math>
by Cauchy-Schwarz. Thus at least <math>1/3-O(1/n)</math> of the pairs are monochromatic. Taking expectations and using the pigeonhole principle, we conclude that one of the distances has a probability at least <math>1/3 -O(1/n)</math> of being monochromatic, giving the claim. <math>\Box</math>

=== Lemma 38 ===

Let ABC be a unit-edge equilateral triangle, and let D be an arbitrary point. Let <math>|AD|, |BD|</math> and <math>|CD|</math> be <math>d,e,f</math> respectively. Then <math>p(d)+p(e)+p(f) \leq 1</math>. In particular, examining the case <math>e=f</math>, if <math>p(d) \geq k</math> then <math>p(\sqrt(d(d \pm \sqrt 3) + 1) \leq (1-k)/2</math>.

'''Proof''' At most one of <math>AD, BD</math> and <math>CD</math> can be monochromatic. <math>\Box</math>

Note: A consequence is that a 4-chromatic unit-distance graph G can demonstrate CNP <math>> 4</math> if, for the {d,e,f} arising from some choice of D above, G contains three equal-sized non-empty sets v_d, v_e, v_f of vertex-pairs such that (a) each vertex-pair within v_d is at distance d (resp. e and f), and (b) in any 4-colouring of G, more than 1/3 of the vertex-pairs in the union of the three sets are monochromatic. Note that this demonstration does not require that v_d contain all the vertex-pairs of G that are at distance d (resp. e and f), nor even that the graph {A,B,C,D} which gives rise to {d,e,f} be a subgraph of G. It seems plausible to find such a graph that is small (and/or symmetrical) enough that its colourings can be human-analysed to establish this property.

== Simplification rules for triplets of points in the complex plane ==
Deduced from the rule <math>{\bf P}(A\land B)+{\bf P}(A\land \lnot B)={\bf P}(A)</math>.
:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) = {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) - {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) ) - {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) \neq {\mathbf c}(z_0) ) + {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) = {\mathbf c}(z_0) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

== Proofs of bounds for conditional probabilities ==
The trivial case, valid where <math>\left|d\right|\neq 1</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) = {\mathbf c}(d) )=0</math>

Trivial plus Baye's Theorem, valid where <math>d\neq 0</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) )=\frac{{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )}\leq 1</math>
Rearrange:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )+{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1</math>

Spindle method: for <math>\left|d\right|=d\geq 1/2</math> and <math>\theta=2\text{arcsin}\left(\frac{1}{2d}\right)</math>:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{i\theta}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) ) = \frac{1}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )} - 1\leq 1</math>
which is another way to see <math>\left|d\right|=d\geq 1/2\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1/2</math>.

== Further questions ==
* For <math>n,m\geq CNP</math>, what consistent relationships exist between <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert n\text{ colors}\right)</math> and <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert m\text{ colors}\right)</math>? How can these relationships be used to sharpen arguments of the probabilistic formulation?

[[Category:Polymath16]]

Hadwiger-Nelson problem

2018-07-31T20:54:20Z

Aubrey: /* Order of a k-chromatic unit-distance graph in the plane */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| '''553'''
| 2722
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5281 Every unit distance graph with at most 6906 vertices is 6-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Hadwiger-Nelson problem

2018-07-31T14:51:21Z

Aubrey: /* Order of a k-chromatic unit-distance graph in the plane */

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds <math>4 \leq CNP \leq 7</math> are classical; recently [deG2018] it was shown that <math>CNP \geq 5</math>. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The '''Polymath16''' project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

* '''Goal 1''': Find progressively smaller 5-chromatic unit-distance graphs.
* '''Goal 2''': Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
* '''Goal 3''': Apply these simpler graphs to inform progress in related areas. For example:
** Find a 6-chromatic unit-distance graph in the plane.
** Improve the corresponding bound in higher dimensions.
** Improve the current record of 383/102 for the fractional chromatic number of the plane.
** Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

== Polymath threads ==

* [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/ Polymath proposal: finding simpler unit distance graphs of chromatic number 5], Aubrey de Grey, Apr 10 2018. ('''Active discussion thread''')
* [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/ Polymath16, first thread: Simplifying de Grey’s graph], Dustin Mixon, Apr 14, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic Polymath16, second thread: What does it take to be 5-chromatic?], Dustin Mixon, Apr 22, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/ Polymath16, third thread: Is 6-chromatic within reach?], Dustin Mixon, May 1, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/ Polymath16, fourth thread: Applying the probabilistic method], Dustin Mixon, May 5, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/ Polymath16, fifth thread: Human-verifiable proofs], Dustin Mixon, May 10, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/05/29/polymath16-sixth-thread-wrestling-with-infinite-graphs/ Polymath16, sixth thread: Wrestling with infinite graphs], Dustin Mixon, May 29, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/ Polymath16, seventh thread: Upper bounds], Dustin Mixon, June 16, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/06/24/polymath16-eighth-thread-more-upper-bounds/ Polymath16, eighth thread: More upper bounds], Dustin Mixon, June 24, 2018. (''Inactive research thread'')
* [https://dustingmixon.wordpress.com/2018/07/02/polymath16-ninth-thread-searching-for-a-6-coloring/ Polymath16, ninth thread: Searching for a 6-coloring], Dustin Mixon, July 2, 2018. ('''Active research thread''')

== Notable unit distance graphs ==

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for <math>CNP</math>; in particular, if one can find a unit distance graph with no 4-colorings, then <math>CNP \geq 5</math>. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

<math>G_1 \oplus G_2</math> denotes the Minkowski sum of two unit distance graphs <math>G_1,G_2</math> (vertices in <math>G_1 \oplus G_2</math> are sums of the vertices of <math>G_1,G_2</math>). <math>G_1 \cup G_2</math> denotes the union. <math>\mathrm{rot}(G, \theta)</math> denotes <math>G</math> rotated counterclockwise by <math>\theta</math>. <math>\mathrm{trim}(G,r)</math> denotes the trimming of <math>G</math> after removing all vertices of distance greater than <math>r</math> from the origin.

Another basic operation is '''spindling''': taking two copies of a graph <math>G</math>, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus <math>\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))</math> for various natural numbers <math>t</math>, particularly the [https://oeis.org/A003136 Loeschian numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, and the distances <math>\sqrt{t}</math> are the distances that arise in the triangular lattice. The rings <math>R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]</math>, where <math>t_1,t_2,\dots</math> are the Loeschian numbers, seem particularly relevant, thus <math>R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]</math>, etc.. Closely related rings are the rings <math>\overline{R_n}</math> generated by the unit vectors in <math>R_n</math> and their inverses.

Note that the square root <math>\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )</math> of <math>\omega_3</math> lies in <math>R_2</math>, thanks to the identity
:<math> \eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).</math>

{| border=1
|-
!Name!!Number of vertices!! Number of edges !! Structure !! Group !! Colorings
|-
| [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]
| 7
| 11
| Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [http://mathworld.wolfram.com/GolombGraph.html Golomb graph]
| 10
| 18
| Contains the center and vertices of a hexagon and equilateral triangle
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| Not 3-colorable
|-
| [https://arxiv.org/abs/1804.02385 H]
| 7
| 12
| Vertices and center of a hexagon
| <math>{\bf Z}[\omega_1]</math>
| Has essentially four 4-colorings, two of which contain a monochromatic <math>\sqrt{3}</math>-triangle. Every 5-coloring has a monochrome <math>\sqrt{3}</math>-edge or a monochrome <math>2</math>-edge
|-
| [https://arxiv.org/abs/1804.02385 J]
| 31
| 72
| Contains 13 copies of H
| <math>{\bf Z}[\omega_1]</math>
| Has essentially six 4-colorings in which no H has a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 K]
| 61
| 150
| Contains 2 copies of J
|
| In all 4-colorings lacking an H with a monochromatic <math>\sqrt{3}</math>-triangle, all pairs of vertices at distance 4 are monochromatic
|-
| [https://arxiv.org/abs/1804.02385 L]
| 121
| 301
| Contains two copies of K and 52 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ <math>L_1</math>]
| 97
|
| Has 40 copies of H
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120295 <math>L_2</math>]
| 120
| 354
|
|
| All 4-colorings contain an H with a monochromatic <math>\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1804.02385 T]
| 9
| 15
| Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
|
|
|-
| [https://arxiv.org/abs/1804.02385 U]
| 15
| 33
| Three copies of T at 120-degree rotations: <math>T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 V]
| 31
| 30
| <math>\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>V_1</math>]
| 61
| 60
| Union of V and a rotation of V: <math>V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))</math>
| <math>{\bf Z}[\omega_1,\omega_3, \omega_4]</math>
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_b</math>]
| 25
| 24
| Star graph
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_x</math>]
| 13
| 12
| Subgraph of <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_z</math>]
|
|
| Subgraph of <math>V_a</math>; shares a line of symmetry with <math>V_a</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_y</math>]
| 13
| 12
| Subgraph of <math>V_b</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A</math>]
| 37
| 36
| Unit vectors with angles <math>i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}</math>
|
|
|-
| [https://arxiv.org/abs/1804.02385 W]
| 301
| 1230
| Cartesian product of V with itself, minus vertices at more than <math>\sqrt{3}</math> from the centre (i.e. <math>\mathrm{trim}(V \oplus V, \sqrt{3})</math>)
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>W_1</math>]
|
|
| Trimmed product of V with itself (<math>\mathrm{trim}(V \oplus V, 1.95)</math>)
|
|
|-
| [https://arxiv.org/abs/1804.02385 M]
| 1345
| 8268
| Cartesian product of W and H (<math>W \oplus H</math>)
| <math>{\bf Z}[\omega_1, \omega_3]</math>
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120274 <math>M_1</math>]
| 278
|
| Deleting vertices from M while maintaining its restriction on H
|
| No 4-colorings have a monochromatic triangle in the central copy of H
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3933 <math>M_2</math>]
| 7075
|
| Sum of H with a trimmed product of <math>V_1</math> with itself
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 N]
| 20425
| 151311
| Contains 52 copies of M arranged around the H-copies of L
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_0</math>]
| 1585
| 7909
| N "shrunk" by stepwise deletions and replacements of vertices
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1804.02385 G]
| 1581
| 7877
| Deleting 4 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ <math>G_1</math>]
| 1577
|
| Deleting 8 vertices from <math>G_0</math>
|
| Not 4-colorable
|-
| [https://polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/#comment-120318 <math>G_2</math>]
| 874
| 4461
| Juxtaposing two copies of M and shrinking
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3867 <math>G_3</math>]
| 826
| 4273
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4105 <math>G_4</math>]
| 803
| 4144
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4316 <math>G_5</math>]
| 633
| 3166
| Subgraph of two copies of <math>V \oplus V \oplus V</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4465 <math>G_6</math>]
| 610
| 3000
|
|
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.12181 <math>G_7</math>]
| '''553'''
| 2722
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 R]
|
|
| Union of <math>W_1</math> and a rotated copy of <math>W_1</math>
|
|
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3946 <math>\mathrm{trim}(R \oplus H, 1.67)</math>]
| 2563
|
| Trimmed sum of R and H
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>V \oplus V \oplus H</math>]
|
|
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_{64/9}]</math>
| Has two vertices forced to be the same color in a 4-coloring; also no monochromatic <math>\sqrt{3}</math>-triangles
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4074 <math>G_{745}</math>]
| 745
|
| Subgraph of <math>V \oplus V \oplus H</math>
|
| Has two vertices forced to be the same color in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3085
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3961 <math>V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H</math>]
| 3049
|
|
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3950 <math>G_{1951}</math>]
| 1951
|
| Trimmed version of <math>V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H</math>
|
| Not 4-colorable
|-
| [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-3970 <math>V_A \oplus V_A \oplus V_A</math>]
| 6937
| 44439
|
| <math>{\bf Z}[\omega_1, \omega_3, \omega_4]</math>
| Not 4-colorable
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{40}</math>]
| 40
| 82
|
|
| Any 4-coloring that avoids a monochromatic <math>\sqrt{11/3}</math>-edge has two specific vertices forced to be the same color
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{79}</math>]
| 79
| 165
| Spindling of <math>G_{40}</math>
|
| Any 4-coloring has a monochromatic <math>\sqrt{11/3}</math>-edge
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{49}</math>]
| 49
| 180
|
| <math>{\mathbf Q}[\sqrt{3},\sqrt{11}]^2</math>
| Any 4-coloring either has a specific <math>\sqrt{11/3}</math>-edge monochromatic, or a monochromatic <math>1/\sqrt{3}</math>-triangle
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{51}</math>]
| 51
|
|
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring of plane
|-
| [https://arxiv.org/abs/1805.00157 <math>G_{627}</math>]
| 627
| 2982
| Contains <math>G_{51}</math>
|
| Has a specific <math>1/\sqrt{3}</math>-triangle which cannot be monochromatic in a 4-coloring
|-
| [https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4161 <math>G_{103}</math>]
| 103
|
|
|
| All 4-colorings contain a monochromatic <math>2/\sqrt{3}</math>-edge
|-
| regular pentagon with unit side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}+1)/2</math>-edge
|-
| regular pentagon with <math>(\sqrt{5}-1)/2</math> side length
| 5
| 5
|
|
| All 4-colorings contain a monochromatic <math>(\sqrt{5}-1)/2</math>-edge
|-
| [http://www.hansparshall.com/txt/n7superCoordinates.txt <math>G_{21}</math>]
| 21
| 49
|
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Implies <math>p_2 \geq 1/4</math>
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4391 <math>G_{43}</math>]
| 43
|
| <math>\{0,1,\eta,\overline{\eta}, \eta -\overline{\eta}, \eta - \overline{\eta}\omega_1, 1+\eta \omega_1^2, 1+\overline{\eta\omega_1^2}\} \cdot \langle \omega_1 \rangle</math>
| <math>{\bf Z}[\omega_1,\omega_3]</math>
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4398 <math>G_{24}</math>]
| 24
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4423 <math>G_{26}</math>]
| 26
|
|Except for the origin, all vertices lie on one of 3 unit circles.
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/05/polymath16-fourth-thread-applying-the-probabilistic-method/#comment-4409 <math>G_{34}</math>]
| 34
|
|
|
| Origin cannot be bichromatic
|-
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4420 <math>G_{30}</math>]
| 30
|
|
|
| Origin cannot be bichromatic
|}

== Lower bounds under various criteria ==

=== Order of a k-chromatic unit-distance graph in the plane ===

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable this MO thread] for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization. Progress thus far is as follows:

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-4176 Every unit distance graph with at most 16 vertices is 5-colorable].

[https://dustingmixon.wordpress.com/2018/05/01/polymath16-third-thread-is-6-chromatic-within-reach/#comment-5268 Every unit distance graph with at most 24 vertices is 5-colorable].

=== Tile-based colourings (tilings) ===

Let a tile-based colouring (hereafter a "tiling") be one consisting of monochromatic regions ("tiles"), each of finite area greater than some positive number, and each bounded by a Jordan curve. This class of colourings is of interest because, even though a 5-colouring of the plane has not been ruled out, such a colouring cannot be a tiling (as shown by Townsend [Tow2005], correcting a minor error in an earlier proof by Woodall [W1973]). An independent proof was given by Coulson [C2004]. In [T1999], Thomassen further showed that any tile-based 6-coloring would have to be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

It is, therefore, of interest to determine whether the scaleability criterion relied upon by Thomassen can be removed, i.e. whether a (necessarily unscaleable) tiling can have only six colours.

Denote the annulus-like region consisting of all points at unit distance from some point in a tile T by AE_T, standing for "annulus of exclusion". Admissible tilings of the plane can in principle have cases where two tiles lie inside each other's AE; we know of such tilings that are 7-colourable and are not trivially transformable into tilings lacking such "Siamese tiles". Tiles in a k-colour tiling that features Siamese tiles can in principle have arbitrarily many vertices (as far as we yet know). By contrast, if a k-colour tiling does not feature Siamese tiles, much can be said: no tile can have more than k-1 vertices, and at most k of them can meet at a vertex (or a simple transformation can make this so without making the tiling non-k-colourable). There are, therefore, only finitely many ways to fit such tiles together, which makes it attractive to try to demonstrate computationally that a 6-colour tiling without Siamese tiles cannot exist, especially since we have tentative reasons to hope that tilings that do have Siamese tiles can be proven impossible by other means.

We can begin by enumerating all admissible neighbourhoods of a tile in a 6-colour tiling. Each vertex V of a tile T can have at most three other tiles meeting it, not counting the tiles that share the edges of T that meet at V; call these the vertex-neighbours of T at V. If two vertices of T have vertex-neighbours of the same colour, those vertices must be unit distance apart; similarly, if a vertex-neighbour of T at V is the same colour as an edge-neighbour of T, that edge must be an arc of radius 1 centred at V. This allows us to encode each admissible tile neighbourhood as a list d of valencies (each between 3 and 6) of the tile's n = 2, 3, 4 or 5 vertices (we can ignore the one-vertex case), together with a list S of vertex pairs {v_i,v_j} and vertex-edge pairs {v_i,e_j} that must be unit distance apart. (We index edges such that e_i joins v_i and v_i+1.) The combination {d,S} must then obey a number of other constraints:

# Edges at unit distance from a point include their ends: thus, if {v_i,e_j} is in S, {v_i,v_j} and {v_i,v_j+1} must also be.
# A vertex cannot be at distance 1 from itself: thus, given (1), neither {v_i,e_i} nor {v_i,e_i-1} can be in S.
# The diameter of neighbouring tiles cannot exceed 1: thus, if d_i = 3, at least one of {v_i-1,v_i} and {v_i,v_i+1} must not be in S.
# Quadrilaterals ABCD with AB=CD=1 must have at least one of AC,AD,BC,BD longer than 1: thus, no disjoint pair {v_i,v_i+1} and {v_j,v_k} can both be in S.
# Six tiles meeting at a vertex must cover a unit-radius disk: thus,
#* if n=4 or 5, at most one d_i can be 6, and if n=2, neither can.
#* if d_i = 6, {v_i-1,v_i+1} must be in S.
# Pigeonhole principle wrt any two vertices: for all i,j, if d_i + d_j + min(|j-i|,n-|j-i|,n-2) >= 10, {v_i,v_j} must be in S.
# Pigeonhole principle wrt a vertex and the tile's edges: for all i, at least d_i + n-8 of the {v_i,e_j} must be in S.
# Pigeonhole principle wrt all edges and vertices combined: If d = {4,4,4} or some d_i = d_i+1 = 4 then S must include a {v_i,e_j}.

We are working to enumerate all cases that satisfy this list of constraints. Once that is done, we will examine which pairs (and beyond) of admissible tiles can be juxtaposed. The hope is that all cases will run into irreconcileable conflicts at a manageable radius from an intial tile; this is based on our failure thus far to 6-tile a disk of radius greater than slightly over 2.

=== Colourings that are not tile-based ===

Intuitively, a colouring of the plane using the minimum number of colours will surely be a tiling. However, this is not necessarily so, and indeed the possibility that a non-tile-based 5-colouring of the plane exists has not been ruled out. However, no example has yet been found (to our knowledge) of a non-tile-based partitioning of the plane that cannot trivially be transformed into a tiling without increasing its chromatic number. Do such partitionings exist? If they could be proved not to, the chromatic number of the plane would be lower-bounded at 6 without the discovery of a 6-chromatic finite unit-distance graph.

An intermediate case that may be worth exploring (but we have not yet done so) is that of partitionings in which each point is part of a monochromatic region of positive area bounded by a Jordan curve, but in which arbitrarily small such regions are present. The proofs mentioned above of a lower bound of 6 rely on the existence of a positive minimum area for a tile.

== Virtual edge ==

''Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length <math>d</math> and chromatic number <math>4</math>.''

A '''virtual edge''' of a unit distance graph <math>G</math> is a distance <math>d</math> with the property that every 4-coloring of <math>G</math> contains a monochromatic pair of vertices of distance exactly <math>d</math>. Observe that if a unit distance graph <math>G</math> has a virtual edge at distance <math>d</math>, and if there is another unit distance graph <math>H</math> with a pair of vertices at distance <math>d</math> that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of <math>H</math> to every virtual edge in <math>G</math>.

Known examples of virtual edges include:

* <math>G_{40}</math> has a virtual edge at distance <math>\sqrt{11/3}</math>.
* <math>V \oplus V \oplus H</math> has a (single) virtual edge at distance <math>8/3</math>.
* [https://dustingmixon.wordpress.com/2018/04/22/polymath16-second-thread-what-does-it-take-to-be-5-chromatic/#comment-4117 <math>(\sqrt{3}\pm 1)/\sqrt{2}</math> are virtual edges of some graphs].

== Bichromatic virtual edge ==

===Definition===
If a unit distance graph <math>H</math> exists with a specific pair of vertices which are distance <math>d</math> apart and is bichromatic in all proper <math>n</math>-colorings of <math>H</math>, then we say that <math>H</math> virtualizes a '''bichromatic virtual edge''' with length <math>d</math> and chromatic number <math>n</math>.

===Vacuous properties===
Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number <math>CNP-1</math>.

If a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n</math> exists, then a bichromatic virtual edge with length <math>d</math> and chromatic number <math>n-1</math> exists.

===Convenience and devirtualization===
Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph <math>G</math> with graphs which virtualize the bichromatic virtual edges through '''clamping''' produces a unit distance graph <math>G'</math> where <math>\chi(G')\geq\chi(G)</math>.

Devirtualizing bichromatic virtual edges of a graph <math>G</math> (i.e. clamping) has no benefit unless nontrivial points of <math>G</math> and <math>H_0</math> overlap or nontrivial points of <math>H_1</math> and <math>H_0</math> overlap, where <math>H_0</math> and <math>H_1</math> are graphs virtualizing bichromatic virtual edges of <math>G</math>. Such overlaps may cause <math>\chi(G')>\chi(G)</math>. If no nontrivial overlaps exist, then <math>\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)</math>.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

===Relation to rings===
A '''devirtualized ring''' of graph <math>G</math> is a ring which contains all the vertices of a devirtualization of <math>G</math>, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets <math>\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]</math>.

Let the vertices of graph <math>G</math> generate the ring <math>S_0[T_0]</math>.
Let the bichromatic virtual edges be virtualizable by the graphs <math>H_1,\dots,H_m</math>.
Let <math>S_k[T_k]</math> be the ring generated by the vertices of <math>H_k</math>.
Let <math>S</math> be the ring generated by <math>\bigcup_{k=0}^m S_k</math>.
Let <math>T=\bigcup_{k=0}^m T_k</math>.
Then <math>S[T]</math> is a devirtualized ring of <math>G</math>.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set <math>T</math>.

===Multiplicative property===
Let <math>H_0</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_0</math> and chromatic number <math>n</math>.
Let <math>H_1</math> be a graph virtualizing a bichromatic virtual edge with length <math>d_1</math> and chromatic number <math>n</math>.
Create a graph <math>H_2</math> by scaling <math>H_0</math> by a factor of <math>d_1</math>. Graph <math>H_2</math> then virtualizes a bichromatic virtual edge length <math>d_0d_1</math> and chromatic number <math>n</math>.

===Notable bichromatic virtual edges===
{| border=1
|-
! Chromatic number !! Length <math>d</math>!! Proof
|-
| any
| 1
| trivial case
|-
| 2
| <math>0\leq d\leq 3</math>
| moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
|-
| 3
| <math>\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}</math>
| equilateral triangle tiling
|}

===Continuous ranges of bichromatic virtual edges===
Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number <math>n</math> without having a specific pair which is monochrome for every <math>n</math>-coloring.
Let the range of lengths be on the interval <math>d_0\leq d\leq d_1</math>.

If <math>1 < d_1</math>, then let <math>k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all lengths larger than <math>d_0^k</math>. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies <math>CNP> n</math>.

If <math>d_0 < 1</math>, then let <math>k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k</math>. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number <math>n</math> for all non-zero lengths smaller than <math>d_1^k</math>. Considering a set of <math>CNP+1</math> points all within distance <math>d_1^k</math> of each other implies <math>CNP> n</math>.

Using a flexible bichromatic virtual edge of chromatic number <math>n</math> and a graph <math>H</math> of chromatic number <math>n+1</math>, a flexible a graph <math>H'</math> of chromatic number <math>n+1</math> can be created by scaling <math>H</math> to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number <math>n</math> as a replacement of each rigid bichromatic virtual edge of <math>H</math>.

<math>H</math> virtualizing a bichromatic virtual edge of chromatic number <math>n+1</math> does not necessarily mean the graph <math>H'</math> virtualizes a bichromatic virtual edge.

==Best known results for the chromatic number in higher dimensions==

{| border=1
|-
! Space !! lower bound on CN !! Number of vertices !! Number of edges !! Upper bound on CN
|-
| <math>\mathbb{R}^1</math>
| 2
| 2
| 1
| 2
|-
| <math>\mathbb{R}^2</math>
| [https://arxiv.org/abs/1804.02385 5]
| [https://arxiv.org/abs/1805.12181 553]
| 2722
| 7
|-
| <math>\mathbb{R}^3</math>
| [https://www.sciencedirect.com/science/article/pii/S0012365X00004064 6]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 59]
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 183]
| [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.684&rep=rep1&type=pdf 15]
|-
| <math>\mathbb{R}^4</math>
| [https://s3.amazonaws.com/academia.edu.documents/34568816/r4_march_22_revised.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1526311039&Signature=%2FrBgIHVOjapKNjsPiizHVO3IpJQ%3D&response-content-disposition=inline%3B%20filename%3DOn_the_Chromatic_Number_of_R.pdf 9]
| 65
| 588
| [https://link.springer.com/chapter/10.1007/978-3-642-55566-4_32 54]
|-
| <math>\mathbb{R}^5</math>
| [https://www.sciencedirect.com/science/article/pii/S0097316596800069 9]
|
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4463 156]
|-
| <math>\mathbb{R}^6</math>
| [https://arxiv.org/abs/1408.2002 12]
| 175
|
| [https://dustingmixon.wordpress.com/2018/05/10/polymath16-fifth-thread-human-verifiable-proofs/#comment-4583 462]
|-
| <math>\mathbb{R}^7</math>
| [https://arxiv.org/abs/1408.2002 16]
| 168
| 4396
|
|-
| <math>\mathbb{R}^8</math>
| [https://arxiv.org/abs/1409.1278 19]
| 289
|
|
|-
| <math>\mathbb{R}^9</math>
| [https://arxiv.org/abs/1512.03472 22]
| 672
|
|
|-
| <math>\mathbb{R}^{10}</math>
| [https://arxiv.org/abs/1512.03472 30]
| 960
|
|
|-
| <math>\mathbb{R}^{11}</math>
| [https://arxiv.org/abs/1512.03472 35]
| 1320
|
|
|-
| <math>\mathbb{R}^{12}</math>
| [https://arxiv.org/abs/1512.03472 37]
| 1760
|
|
|}

== Probabilistic formulation ==

See [[Probabilistic formulation of Hadwiger-Nelson problem]].

== Algebraic formulation ==

See [[Algebraic formulation of Hadwiger-Nelson problem]].

== Excluding bichromatic vertices ==

See [[Excluding bichromatic vertices]].

== Coloring <math>R_2</math> ==

See [[Coloring R_2]].

== Further questions ==

* What are the [http://mathworld.wolfram.com/IndependenceRatio.html independence ratios] of the above unit distance graphs?
* What are the [https://en.wikipedia.org/wiki/Fractional_coloring fractional chromatic numbers] of these graphs?
* What are the [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz numbers] of these graphs?
** The Lovasz theta function value of Lovasz number of <math>G_2</math> at the complement [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3844 is in the interval [3.3746, 3.3748]].
* What about the Erdos unit distance graph (<math>n</math> vertices, <math>n^{1+c/\log\log n}</math> edges)?
* Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3899 Varga] and [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3940 Nazgand] provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3900 prove that the measurable chromatic number of the plane is at least 6].
* It appears as though the coordinates of our smallest 5-chromatic graph lie in <math>\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]</math> (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3879 this]). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3914 this]). [https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/#comment-3934 David Speyer suggests] looking at <math>\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]</math> next.
* What is the smallest cardinality of a subset of the plane which contains at least <math>n</math> colours in every colouring of the plane?
* Can the lower bound for <math>CNP\geq 5</math> be extended to all <math>L^p</math> norms where <math>p>1</math>, similar to how the Moser spindle was generalized?

== Blog, forums, and media ==

* [https://rjlipton.wordpress.com/2011/05/22/more-on-coloring-the-plane/ More On Coloring The Plane], Richard Lipton, May 22, 2011.
* [https://mathoverflow.net/questions/236392/has-there-been-a-computer-search-for-a-5-chromatic-unit-distance-graph Has there been a computer search for a 5-chromatic unit distance graph?], Juno, Apr 16, 2016.
* [https://quomodocumque.wordpress.com/2018/04/09/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Jordan Ellenberg, Apr 9 2018.
* [https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the-chromatic-number-of-the-plane-is-at-least-5/ Aubrey de Grey: The chromatic number of the plane is at least 5], Gil Kalai, Apr 10 2018.
* [https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Dustin Mixon, Apr 10, 2018.
* [https://www.scottaaronson.com/blog/?p=3697 Amazing progress on long-standing problems], Scott Aaronson, Apr 11 2018.
* [https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/ The chromatic number of the plane is at least 5, Part II], Dustin Mixon, Apr 13 2018.
* [http://community.wolfram.com/groups/-/m/t/1320004 A 5-chromatic unit distance graph], Ed Pegg, Apr 13 2018.
* [http://aperiodical.com/2018/04/the-chromatic-number-of-the-plane-is-at-least-5/ The chromatic number of the plane is at least 5], Katie Steckles, Apr 17 2018.
* [https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/ Decades-Old Graph Problem Yields to Amateur Mathematician], Evelyn Lamb, Quanta, Apr 17, 2018.
* [https://www.heise.de/newsticker/meldung/Zahlen-bitte-Wie-bunt-ist-die-Ebene-4024574.html Zahlen, bitte! 5 - Wie bunt ist die Ebene?], Harald Bögeholz, Heise, Apr 17, 2018.
* [http://www.sciencemag.org/news/2018/04/amateur-mathematician-cracks-decades-old-math-problem Amateur mathematician cracks decades-old math problem], Katie Langin, Science News, Apr 18, 2018.
* [https://mathoverflow.net/questions/298198/how-much-of-the-plane-is-4-colorable How much of the plane is 4-colorable?], Dustin Mixon, Apr 18, 2018.
* [https://www.sciencealert.com/amateur-solves-decades-old-maths-problem-about-colours-that-can-never-touch-hadwiger-nelson-problem An Amateur Solved a 60-Year-Old Maths Problem About Colours That Can Never Touch], Peter Dockrill, ScienceAlert, Apr 19, 2018.
* [https://www.zmescience.com/science/math/amateur-mathematician-solves-problem-20042018/ Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem], Mihai Andrei, ZME Science, Apr 20, 2018.
* [https://automaths.blog/2018/04/21/5-nuances-daubrey-de-grey/ 5 nuances d’Aubrey de Grey], Automaths, Apr 21, 2018.
* [https://science.howstuffworks.com/math-concepts/amateur-solves-part-of-decades-old-math-problem.htm Amateur Solves Part of Decades-Old Math Problem], HowStuffWorks, Apr 30, 2018.
* [https://www.nemokennislink.nl/publicaties/het-platte-vlak-heeft-minstens-vijf-kleuren-nodig/ Het platte vlak heeft minstens vijf kleuren nodig], Kennislink, May 3, 2018.
* [https://index.hu/tudomany/2018/05/04/egy_biologus_oldott_meg_egy_olyan_problemat_amire_a_matematikusok_mar_60_eve_keptelenek/ Egy biológus oldott meg egy olyan problémát, amire a matematikusok már 60 éve képtelenek], Index.hu, May 4, 2018.

== Code and data ==

[https://www.dropbox.com/sh/ufknm1v9gtbhad3/AACB2xwaXYx5EGda38_L-0foa?dl=0 This dropbox folder] will contain most of the data and images for the project.

Data:

* [https://www.dropbox.com/s/x6kvu7b3wvdvqsn/graph.dimacs?dl=0 The 1585-vertex graph in DIMACS format]
* [https://www.dropbox.com/s/qk3gbpjvvsjfsc3/sat.dimacs?dl=0 A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format]
* [https://www.dropbox.com/s/nipqikfzcfn9o5a/vertices.sage?dl=0 The vertices of this graph in explicit Sage notation]
* The graph <math>G_2</math>: [http://www.cs.utexas.edu/~marijn/CNP/874.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/874.edge edges (DIMACS)]
* The graph <math>G_3</math>: [http://www.cs.utexas.edu/~marijn/CNP/826.vtx vertices (Mathematica)] [http://www.cs.utexas.edu/~marijn/CNP/826.edge edges (DIMACS)] [http://www.cs.utexas.edu/~marijn/CNP/826.pdf Visualization]
* The [https://www.dropbox.com/sh/ufknm1v9gtbhad3/AADfw9WO1ol2ayQNJEtGf8oBa/Graphs?dl=0 densest unit-distance graphs] on an <math>n\times n</math> grid for <math>n=10,20,\ldots,100</math> (DIMACS format).

Code:

* [https://www.dropbox.com/s/6u1jctbjy38t383/lovaszmoser.m?dl=0 MATLAB script for computing Lovasz number]
* [https://files.jixco.de/pm16/zzvtx/ Python code for converting a list of vertices in Mathematica format into vertices in Z^n]
* [https://files.jixco.de/pm16/edge2cnf.py Python code for converting a DIMACS edge list into a CNF formula (forcing up to 3 suitable vertices to a fixed color, to break some symmetries)]

Software:

* [http://cvxr.com/cvx/ CVX]
* [http://www.labri.fr/perso/lsimon/glucose/ Glucose 4.0]
* [http://minisat.se/ Minisat]

== Wikipedia ==

* [https://en.wikipedia.org/wiki/Graph_coloring#Chromatic_number Chromatic number]
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory) de Bruijn-Erdos theorem]
* [https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]
* [https://en.wikipedia.org/wiki/Lov%C3%A1sz_number Lovasz number]
* [https://en.wikipedia.org/wiki/Moser_spindle Moser spindle]

== Bibliography ==
* [B2008] B. Bukh, [https://doi.org/10.1007/s00039-008-0673-8 Measurable sets with excluded distances], Geometric and Functional Analysis 18 (2008), 668-697.
* [C2004] D. Coulson, On the chromatic number of plane tilings, J. Aust. Math. Soc. 77 (2004), 191–196.
* [CR2015] D. Cranston, L. Rabern, [https://arxiv.org/abs/1501.01647 The fractional chromatic number of the plane], arXiv:1501.01647
* [deBE1951] N. G. de Bruijn, P. Erdős, [http://www.math-inst.hu/~p_erdos/1951-01.pdf A colour problem for infinite graphs and a problem in the theory of relations], Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.
* [deG2018] A. de Grey, [https://arxiv.org/abs/1804.02385 The chromatic number of the plane is at least 5], arXiv:1804.02385
* [EI2018] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.00157 The chromatic number of the plane is at least 5 - a new proof], arXiv:1805.00157
* [EI2018b] G. Exoo, D. Ismailescu, [https://arxiv.org/abs/1805.06055 The Hadwiger-Nelson problem with two forbidden distances], arXiv:1805.06055
* [F1981] K.J. Falconer, The Realization of distances in measurable subsets covering Rn, J. Combin. Theory Ser. A 31 (1981) 184–189.
* [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
* [H2018] M. Heule, [https://arxiv.org/abs/1805.12181 Computing Small Unit-Distance Graphs with Chromatic Number 5], arXiv:1805.12181
* [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
* [P1998] D. Pritikin, [https://www.sciencedirect.com/science/article/pii/S0095895698918196 All unit-distance graphs of order 6197 are 6-colorable], Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
* [S2009] M. Shinohara, [https://www.sciencedirect.com/science/article/pii/S019566980300218X Classification of three-distance sets in two dimensional Euclidean space], European Journal of Combinatorics Volume 25, Issue 7, October 2004, Pages 1039-1058.
* [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.
* [T1999] C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly 106 (1999) 850-853.
* [Tow2005] Townsend, S.P., Colouring the plane with no monochrome unit. Geombinatorics XIV(4) (2005), 181-193.
* [W1973] Woodall, D.R., Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14 (1973), 187-200.

[[Category:Polymath16]]

Probabilistic formulation of Hadwiger-Nelson problem

2018-07-25T14:33:35Z

Aubrey: /* Lemma 38 */

Suppose for sake of contradiction that we have a 4-coloring <math>c: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with no unit edges monochromatic, thus

:<math>c(z) \neq c(w) \hbox{ whenever } |z-w| = 1. \quad (1)</math>

We can create further such colorings by composing <math>c</math> on the left with a permutation <math>\sigma \in S_4</math> on the left, and with the (inverse of) a Euclidean isometry <math>T \in E(2)</math> on the right, thus creating a new coloring <math>\sigma \circ c \circ T^{-1}: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with the same property. This is an action of the solvable group <math>S_4 \times E(2)</math>.

It is a fact that all solvable groups (viewed as discrete groups) are [https://en.wikipedia.org/wiki/Amenable_group amenable], so in particular <math>S_4 \times E(2)</math> is amenable. This means that there is a finitely additive probability measure <math>\mu</math> on <math>S_4 \times E(2)</math> (with all subsets of this group measurable), which is left-invariant: <math>\mu(gE) = \mu(E)</math> for all <math>g \in S_4 \times E(2)</math> and <math>E \subset S_4 \times E(2)</math>. This gives <math>S_4 \times E(2)</math> the structure of a finitely additive probability space. We can then define a random coloring <math>{\mathbf c}: {\bf C} \to \{1,2,3,4\}</math> by defining <math>{\mathbf c} := {\mathbf \sigma} \circ c \circ {\mathbf T}^{-1}</math>, where <math>({\mathbf \sigma},{\mathbf T})</math> is the element of the sample space <math>S_4 \times E(2)</math>. Thus for any complex number <math>z</math>, the random color <math>{\mathbf c}(z)</math> is a random variable taking values in <math>\{1,2,3,4\}</math>. The left-invariance of the measure implies that for any <math>(\sigma,T) \in S_4 \times E(2)</math>, the coloring <math> \sigma \circ {\mathbf c} \circ T^{-1}</math> has the same law as <math>{\mathbf c}</math>. This gives the color permutation invariance

:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(z_1) = \sigma(c_1), \dots, {\mathbf c}(z_k) = \sigma(c_k) )\quad (2)</math>

for any <math>z_1,\dots,z_k \in {\bf C}</math>, <math>c_1,\dots,c_k \in \{1,2,3,4\}</math>, and <math>\sigma \in S_4</math>, and the Euclidean isometry invariance
:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(T(z_1)) = c_1, \dots, {\mathbf c}(T(z_k)) = c_k. \quad (3)</math>
(In probabilistic language, this means that the random coloring is a [https://en.wikipedia.org/wiki/Stationary_process stationary process] with respect to the action of <math>S_4 \times E(2)</math>. The extraction of a stationary process from a deterministic object is an example of the ''Furstenberg correspondence principle''.)

== Bounds on <math>p_d</math> for 4-colourings ==

A class of correlations that is of particular interest is that of vertex pairs at some distance <math>d</math>. Accordingly, define
:<math>p_d := {\bf P}( \mathbf{c}(0) = \mathbf{c}(d) ).</math>

{| border=1
|-
! distance !! Lower bound !! Lower-bounding graph/method !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math> \geq 1/2</math>
|
|
| 1/2
| Spindle
|
|-
| <math>\geq \frac{2}{\sqrt{15}}</math>
|
|
| <math>\frac{1}{2} - \frac{1}{62}</math>
| Proposition 36
|
|-
| <math>\geq \frac{\sqrt{3}-1}{\sqrt{2}}</math>
|
|
| <math>0.48</math>
| Proposition 36
|
|-
| large enough
|
|
| <math>\frac{323}{675}=0.4785\ldots</math>
| Proposition 36
|
|-
| <math> d\geq 1/n, n>1</math>
|
|
| <math>1-\frac{1}{n}</math>
| (n+1)-gon with one edge length 1 and the rest d, Lemma 34
|
|-
| <math> d\geq 1/(n \sqrt{3}), n>1</math>
|
|
| <math>(3n-2)/3n</math>
| (n+1)-gon with one edge length <math>1/\sqrt{3}</math> and the rest d, Lemma 34
| Not better than the above on intervals <math>\left(\frac{1}{7},\frac{1}{4\sqrt{3}}\right),\left(\frac{1}{4},\frac{1}{2\sqrt{3}}\right)</math>
|-
| 1
| 0
| Unit edge
| 0
| Unit edge
|
|-
| <math>\frac{1}{\sqrt{3}}</math>
| 1/28
| Unit diamond plus centres of triangles, together with H, Corollary 16
| 1/3
| Unit triangle plus its centre
|
|-
| 2
| 1/4
| <math>G_{21}</math>
| <math>0.48</math>
| Proposition 36
| Lower bound computer verified
|-
| <math>{\sqrt{3}}</math>
| 0.26
| H, Prop 36
| <math>0.48</math>
| Proposition 36
|
|-
| <math>{\sqrt{7}}</math>
|
|
| <math>3/8</math>
| Lemma 38 and Corollary 16
|
|-
| <math>{\frac{\sqrt{5}+1}{2}}</math>
| 1/5
| regular pentagon with unit side length
| 2/5
| regular pentagon with unit side length
|
|-
| <math>{\frac{\sqrt{5}-1}{2}}</math>
| 1/5
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
| 2/5
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
|
|-
| <math>{\sqrt{11/3}}</math>
| 1/118
| <math>G_{40}</math>
| <math>0.48</math>
| Proposition 36
| computer-verified
|-
| 8/3
| 1
| <math>V \oplus V \oplus H</math>
| <math>0.48</math>
| Proposition 36
| computer-verified; leads to contradiction
|-
| <math>\frac{\sqrt{6} \pm \sqrt{2}}{2}</math>
| 1/6
| An arrangement of five vertices
| <math>0.5</math> and <math>0.48</math>
| Proposition 36
|
|-
| <math>\sqrt{2}</math>
| 1/14
| A graph of 13 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>\frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math>
| 1/28
| A graph of 13 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>\sqrt{3/2 + \sqrt{33}/6}</math>
| 1/196
| A graph of 9 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>\sqrt{5/3}</math>
| 1/756
| A graph of 33 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>2/\sqrt{3}</math>
| 1/177
| A graph of 103 vertices
| <math>0.48</math>
| Proposition 36
| Computer-verified
|-
| <math>\frac{\sqrt{33} \pm 1}{2\sqrt{3}}</math>
| 1/2
| <math>G_{420}</math>
| <math>0.48</math>
| Proposition 36
| Computer-verified
|}

== Bounds on <math>{\bf P}( \mathbf{c}(0) = \mathbf{c}(d_1) \mid \mathbf{c}(0) \neq \mathbf{c}(d_0) )</math> for 4-colourings ==

{| border=1
|-
! <math>d_1</math> !! <math>d_0</math> !! Lower bound !! Lower-bounding graph !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math>1+(-1)^{1/3}</math>
| <math>2</math>
| 1/2
| H
|
|
| Equals <math>p_{\sqrt 3}/(1-p_2)</math>
|-
| <math>1+(-1)^{1/3}</math>
| <math>1+(-1)^{-1/3}</math>
|
|
| 1/2
| H
|
|}

== Proofs of bounds ==

One can compute some correlations of the coloring exactly:

=== Lemma 1 ===
Let <math>z,w \in {\bf C}</math> with <math>|z-w|=1</math>. Then
:<math> {\bf P}( \mathbf{c}(z) = c ) = \frac{1}{4}\quad (4)</math>
for all <math>c=1,\dots,4</math>,
:<math> {\bf P}( \mathbf{c}(z) = \mathbf{c}(w) ) = 0\quad (5),</math>
and
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c' ) = \frac{1}{12} \quad (6)</math>
for any distinct <math>c,c' \in \{1,2,3,4\}</math>. If <math>u</math> is at a unit distance from both z and w, then
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c'; \mathbf{c}(u) = c'' ) = \frac{1}{24} \quad (6')</math>

'''Proof''' By color invariance (2), the four probabilities in (4) are equal and sum to 1, giving (4). The claim (5) is immediate from (1). From (5) and color invariance, the 12 probabilities in (6) are equal and sum to 1, giving (6). The same argument gives (6').<math>\Box</math>

=== Lemma 2 ===
(Spindle argument) Let <math>|d| \geq 1/2</math>. Then
:<math> p_d \leq \frac{1}{2} \quad (7).</math>

'''Proof''' We can find an angle <math>\theta</math> with <math>|de^{i\theta}-d|=1</math>, then <math>\mathbf{c}(de^{i\theta}) \neq \mathbf{c}(d)</math> almost surely. This means that at least one of the events <math>\mathbf{c}(0) = \mathbf{c}(d)</math>, <math>\mathbf{c}(0) = \mathbf{c}(d e^{i\theta})</math> occurs with probability at most 1/2. The claim now follows from isometry invariance (3). <math>\Box</math>

=== Lemma 3 ===
(Using the K graph) We have
:<math>52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) + {\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} ) \geq 1 \quad (8).</math>

'''Proof''' Consider the 61-vertex graph <math>K</math> from [https://arxiv.org/abs/1804.02385 de Grey's paper]. It has 26 (isometric) copies of H, and thus 52 copies of the triangle <math>(1, e^{2\pi i/3}, e^{4\pi i/3})</math>. With probability at least <math>1 - 52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) </math>, none of these triangles are monochromatic. By the argument in that paper, this implies that the three linking diagonals <math>(-2, +2)</math>, <math>(-2 e^{2\pi i/3}, 2e^{2\pi i/3})</math>, <math>(-2 e^{4\pi i/3}, e^{-4\pi i/3})</math> are monochromatic. This gives the claim. <math>\Box</math>

=== Corollary 4 ===
(Existence of monochromatic <math>\sqrt{3}</math>-triangles) We have <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) \geq \frac{1}{104}</math>.

'''Proof''' The probability <math>{\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} )</math> is at most <math>{\bf P}( \mathbf{c}(-2) = \mathbf{c}(2)) = p_4</math>, which by Lemma 2 is at most 1/2. The claim now follows from Lemma 3. <math>\Box</math>

=== Computer-verified Claim 5 ===
(Using the graph M) One has <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = 0</math> (Note this contradicts Corollary 4).

'''Proof''' This simply reflects the fact that there is no 4-coloring of the 1345-vertex graph M from [https://arxiv.org/abs/1804.02385 de Grey's paper] with its central copy of H containing a monochromatic triangle. One can use other graphs for this purpose, such as the 278-vertex graph <math>M_1</math> or <math>V \oplus V \oplus H</math>. <math>\Box</math>

=== Computer-verified Claim 6 ===
(Using the graph <math>V \oplus V \oplus H</math>) One has <math>p_{8/3} = 1</math> (note this contradicts Lemma 2).

'''Proof''' This reflects the fact that every 4-coloring of <math>V \oplus V \oplus H</math> must assign the same color to 0 and 8/3. There is also a 745-vertex subgraph of <math>V \oplus V \oplus H</math> with the same property. <math>\Box</math>

=== Lemma 7 ===
(Using <math>G_{40}</math>) We have
:<math>59 p_{\sqrt{11/3}} + p_{8/3} \geq 1</math>.

'''Proof''' This reflects the fact that every 4-coloring of the 40-vertex graph <math>G_{40}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which none of the 59 pairs of vertices at distance <math>\sqrt{11/3}</math> apart, will assign the same color to 0 and 8/3. (This is presumably human-verifiable.) <math>\Box</math>

=== Corollary 8 ===
One has
:<math> p_{\sqrt{11/3}} \geq \frac{1}{118}</math>.

'''Proof''' Combine Lemma 7 and Lemma 2. <math>\Box</math>

=== Lemma 9 ===
(Using <math>G_{49}</math>) One has
:<math>18 {\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq p_{\sqrt{11/3}} .</math>

'''Proof''' This reflects the fact that every 4-coloring of the 49-vertex graph <math>G_{49}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which 0 and <math>\sqrt{11/3}</math> have the same color, at least one of the 18 copies of <math>(1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3)</math> is monochromatic. This is potentially human-verifiable. <math>\Box</math>

=== Corollary 10 ===
(Existence of monochromatic <math>1/\sqrt{3}</math>-triangles) One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq \frac{1}{2124}.</math>

'''Proof''' Combine Corollary 8 and Lemma 9. <math>\Box</math>

=== Computer-verified claim 11 ===
One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) = 0</math>. (This contradicts Corollary 10).

'''Proof''' This reflects the fact that the 627-vertex graph <math>G_{627}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] does not have any 4-colorings with <math>1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3</math> monochromatic. <math>\Box</math>

=== Lemma 12 ===
For certain special distances d, one can improve the bound in Lemma 2:

If <math>k \geq 1</math> is a natural number, <math>j\in\mathbb{Z}</math>, <math>\gcd(j,2k+1)=1</math>, and <math>r = \frac{1}{2} \csc\left(\frac{j\pi}{2k+1}\right)</math> then
:<math> p_r \leq \frac{k}{2k+1},</math>
thus for instance
:<math> p_{\frac{1}{\sqrt{3}}} \leq \frac{1}{3}.</math>

'''Proof''' Observe that the regular 2k+1-polygon <math>r, re^{2\pi i/(2k+1)}, r e^{4\pi i/(2k+1)}, \dots, r^{4k\pi i/(k+1)}</math> has unit side lengths. By the pigeonhole principle, we conclude that at most k of these vertices can have the same color as the origin, and the claim follows. <math>\Box</math>

On the other hand, for <math>k=2,j=1</math> we also know from the regular pentagon of unit sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}+1}{2}} \leq \frac{2}{5} \quad (9)</math>
since any 4-coloring of that pentagon has either one or two monochromatic diagonals.

Similarly, for <math>k=2,j=2</math> we also know from the regular pentagon of <math>\frac{\sqrt{5}-1}{2}</math> sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}-1}{2}} \leq \frac{2}{5}</math>
since any 4-coloring of that pentagon has either one or two monochromatic edges. More generally, if <math>a,b,c,d,e</math> are the diagonal lengths of a pentagon with unit sides, then
:<math> 1 \leq p_a + p_b + p_c + p_d + p_e \leq 2.</math>

=== Lemma 13 ===
We have
:<math> 7 p_{\frac{1}{\sqrt{3}}} \geq p_{\sqrt{3}}</math>.

'''Proof''' Consider the unit rhombus <math>0, 1, e^{i\pi/3}, e^{-i\pi/3}</math> together with the centers <math>e^{i\pi/6}/\sqrt{3}, e^{-i\pi/6}/\sqrt{3}</math>. With probability <math>p_{\sqrt{3}}</math>, the two far vertices <math>e^{i\pi/3}, e^{-i\pi/3}</math> are the same color, and then 0,1 will be two other colors. This forces either one of the centers <math>e^{i\pi/6}/\sqrt{3}</math> of a triangle to have a common color with one of the vertices of that triangle, or the two centers must have the same color. Thus in any event one of the seven edges of distance <math>1/\sqrt{3}</math> is monochromatic, giving the claim. <math>\Box</math>

=== Corollary 14 ===
We have <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{728}</math>.

This slightly improves upon the lower bound of 1/2124 coming from Corollary 10.

'''Proof''' Combine Corollary 4 and Lemma 13. <math>\Box</math>

=== Lemma 15 ===
One has
:<math>p_{\sqrt{3}} + p_2 \geq \frac{2}{3}</math> and <math>2 p_{\sqrt{3}} + p_2 \geq 1</math>.

'''Proof''' As noted in de Grey's paper, there are essentially four 4-colorings of H. H has six edges of length <math>\sqrt{3}</math> and three of length <math>2</math>. If we let a denote the number of monochromatic <math>\sqrt{3}</math> edges and b the number of monochromatic <math>2</math>-edges, we see from inspection of all four colorings that <math>(a,b)</math> is either <math>(6, 0), (4,0), (2, 1)</math>, or <math>(0,3)</math>. In particular, one always has <math>\frac{a}{6} + \frac{b}{3} \geq \frac{2}{3}</math> and <math>2\frac{a}{6} + \frac{b}{3} \geq 1</math>. Taking expectations, we obtain the claim. <math>\Box</math>

=== Corollary 16 ===
We have <math>p_2 \geq \frac{1}{6}</math>, <math>p_{\sqrt{3}} \geq \frac{1}{4} </math> and <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{28}</math>.

'''Proof''' Combine Lemma 2, Lemma 15, and Lemma 13. <math>\Box</math>

=== Lemma 17 ===
If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths a,b,c. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also, thus
:<math>{\bf P}(\mathbf{c}(0) \neq \mathbf{c}(a)) + {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(b)) \geq {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(c))</math>
and the claim follows. <math>\Box</math>

Note that Lemma 2 follows from the a=b, c=1 case of this lemma. Iterating this lemma starting with Lemma 2 we can also obtain slightly nontrivial upper bounds on <math>p_a</math> for small values of a, e.g. <math>p_a \leq 1 - 2^{-k}</math> when <math>a \geq 2^{-k}, k\in\mathbb{Z}^+</math>.

Further, we can generalise the a=b case to one in which the triangle is replaced by a (k+1)-gon of which one edge is 1 and the others are all equal, leading to the stronger result <math>p_a \leq 1 - 1/k</math> when <math>a \geq 1/k, k\in\mathbb{Z}^+ \land k>1</math>. Further strengthening is achieved by using <math>1/\sqrt{3}</math> as the long edge, given Lemma 12.

=== Lemma 18 ===
Whenever <math>d>0</math>, one has the inequalities
:<math> |p_{\phi d} - p_d| \leq \frac{2}{5}, p_{\phi d} + p_d \geq \frac{1}{5}, 2p_d - p_{\phi d} \leq 1, 2 p_{\phi d} - p_d \leq 1</math>
where <math>\phi := \frac{1+\sqrt{5}}{2}</math> is the golden ratio. Also we have
:<math> p_{d/\sqrt{3}} \leq \frac{1}{3} + p_d, \frac{1}{2} + \frac{1}{2} p_d.</math>

Note that this generalises (9), as well as a special case of Lemma 12.

'''Proof''' Consider the regular pentagon with sidelength <math>d</math>, so it also has 5 diagonals of length <math>\phi d</math>. Let <math>a \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic edges and let <math>b \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic diagonals. Observe:
* <math>a,b</math> cannot both be zero (pigeonhole principle).
* <math>a</math> cannot be 4. Similarly, <math>b</math> cannot be 4.
* If <math>a=5</math>, then <math>b=5</math>, and conversely.
* If <math>a=0</math>, then <math>b=1,2</math>; similarly, if <math>b=0</math>, then <math>a=1,2</math>.

From this we observe the inequalities
:<math> |\frac{a}{5}-\frac{b}{5}| \leq \frac{2}{5}; \frac{a}{5} + \frac{b}{5} \geq \frac{1}{5}; 2 \frac{a}{5} - \frac{b}{5} \leq 1; 2\frac{b}{5} - \frac{a}{5} \leq 1</math>
and on taking expectations we obtain the first claim. Similarly, if one considers the colorings of an equilateral triangle of sidelength <math>d</math> together with its center, and counts the numbers <math>a,b \in \{0,1,2,3\}</math> of monochromatic edges of length <math>d</math> and <math>d/\sqrt{3}</math> respectively, one observes that one always has <math>\frac{b}{3} \leq \frac{1}{3} + \frac{2}{3} \frac{a}{3}, \frac{1}{2} + \frac{1}{2} \frac{a}{3}</math>, and on taking expectations one obtains the claim.<math>\Box</math>

The hexagon <math>H</math> has essentially four distinct colorings: the coloring <math>\hbox{2tri}</math> with two triangles, the coloring <math>\hbox{1tri}</math> with one triangle, the coloring <math>\hbox{axisym}</math> that is symmetric around an axis, and the coloring <math>\hbox{centralsym}</math> that is symmetric around the central point. This gives four probabilities <math>p_{H = 2tri}, p_{H = 1tri}, p_{H = axisym}, p_{H = centralsym}</math> that sum to 1. By counting the number of monochromatic edges of length <math>\sqrt{3}, 2</math> respectively, one also obtains the identities
:<math>p_{\sqrt{3}} = p_{H = 2tri} + \frac{2}{3} p_{H = 1tri} + \frac{1}{3} p_{H = axisym}; \quad p_2 = \frac{1}{3} p_{H=axisym} + p_{H=centralsym}</math>
which reproves Lemma 15. Also
:<math> {\bf P}( \mathbf{c}(0) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = p_{H = 2tri} + \frac{1}{2} p_{H=1tri}.</math>
Any 4-coloring of L contains at least one triangle within one of its 52 copies of H, thus
:<math> p_{H = 2tri} + \frac{1}{2} p_{H=1tri} \geq \frac{1}{52}</math>
which reproves Corollary 4.

=== Lemma 19 ===
(Hubai) One has <math>p_{H = 1tri} + p_{H = axisym} \geq \frac{1}{10}</math>.

'''Proof''' Consider five copies of H centred at 0,1,2,3,4. With probability at least <math>1 - 5( p_{H = 1tri} + p_{H = axisym} )</math>, none of these copies of H are colored 1tri or axisym, and so must be colored 2tri or centralsym. One can check then that if one of the copies is colored 2tri, then so is any adjacent copy; thus all five copies are colored 2tri, or all five are colored centralsym. In either case we see that -1 and 5 are colored the same color. Comparing with Lemma 2 then gives the claim. <math>\Box</math>

=== Theorem 20 ===
One has <math>p_{H = 1tri} > 0</math>.

'''Proof''' Suppose for contradiction that <math>p_{H = 1tri} = 0</math>. One can then run a version of the de Bruijn-Erdos argument to obtain a coloring in which 1tri hexagons are completely nonexistent (since there are arbitrarily large finite colorings with this property). Consider the triangular lattice <math>{\bf Z}[e^{\pi i/3}]</math>. We 2-color the edges of this lattice by coloring an edge black if it is the short diagonal of a unit rhombus with monochromatic long diagonal, and white otherwise. The four colorings of hexagons lead to four possible colorings at each vertex:

* If H is colored 2tri, then all six edges to the centre of H are black.
* If H is colored 1tri, then two edges to the centre of H at 120 degree angles are white, the other four are black.
* If H is colored axisym, then two opposing edges of the centre of H are black, the other four are white.
* If H is colored centralsym, then all six edges to the centre of H are black.

In particular, as we are assuming no 1tri hexagons, the faces cut out by the black edges have angles 60 degrees, and thus must be equilateral triangles, sectors of angle 60, half-planes, or the entire plane. If there is at least one equilateral triangle, then the rest of the black edges must form an equilateral lattice with that triangle sidelength. This leads to only a small number of possible hexagon colorings in the lattice:

# Case 1: All edges white.
# Case 2: All edges black.
# Case 3.k: For some natural number <math>k \geq 2</math>, the length k edges joining adjacent vertices in some coset of <math>k \cdot {\mathbf Z}[ e^{\pi i/3} ]</math> are all black, and the remaining edges are white.
# Case 4: Each horizontal row consists either entirely of black edges, or entirely of white edges.
# Case 5: Each northwest row consists either entirely of black edges, or entirely of white edges.
# Case 6: Each northeast row consists either entirely of black edges, or entirely of white edges.
# Case 7: Six rays of black edges meeting at a common vertex; all other edges white.

Technically, Case 1 is contained in Cases 4,5,6 as written above, but this will not be an issue. One can view Case 7 as a limiting case <math>k=\infty</math> of Case 3.k; Case 2 is similarly the opposite limiting case <math>k=1</math>.

In the first case, the coloring is periodic with periods <math>2, 2 e^{\pi i/3}</math>. In the second case, it is periodic with periods <math>3, 3 e^{\pi i/3}</math>. In the third case, it is periodic with periods <math>3k, 3k e^{\pi i/3}</math>. Also note that for each k, one can check if Case 3.k holds by inspecting the coloring at a finite number of vertices. Thus the event that Case 3.k holds is "measurable" in the sense that a meaningful probability can be assigned. (But Cases 1,2,4,5,6 are not measurable events, they require an infinite number of points to be inspected, and the probability measure we are using is only finitely additive rather than infinitely additive.) In Case 4, the coloring is periodic with period 2; also, every coset of <math>2 \cdot {\bf Z}[e^{\pi i/3}]</math> is 2-colored. Similarly for Case 5 and 6 (where the periods are <math>2 e^{2\pi i/3}</math> and <math>2 e^{4\pi i/3}</math> respectively.)

Let <math>\alpha_k</math> be the probability that Case 3.k holds for the given value of <math>k</math>. Then <math> \sum_{k=2}^K \alpha_k \leq 1</math> for any k, hence <math>\sum_{k=2}^\infty \alpha_k \leq 1</math>. In particular, we can find <math>K_1</math> such that <math>\sum_{k={K_1}}^\infty \alpha_k \leq 0.1</math> (say). Let <math>P</math> be six times the least common multiple of <math>1,2,\dots,K_1</math>. Then the coloring is P- and <math>P e^{\pi i/3}</math>-periodic for Case 1, Case 2, and all Case 3.k with <math>k \leq K_1</math>. On the other hand, if <math>K_2</math> is sufficiently large depending on <math>P</math>, and Case 3.k holds for some <math>k \geq K_2</math>, then almost all of the hexagons are colored centralsym, which makes the coloring "almost <math>P, P e^{\pi i/3}</math>-periodic" in the sense that
:<math>{\bf c}(z+P e^{\pi i j/3}) = {\bf c}(z) \hbox{ for } j=0,1,2,3,4,5</math>
will hold for at least <math>0.9</math> of the lattice points <math>z \in {\bf Z}[e^{\pi i/3}]</math> with <math>|z| \leq K_2</math>. Similarly for Case 7 (which is sort of a <math>k=\infty</math> limiting case of Case 3.k.) Thus, with the probability <math> \geq 1 - \sum_{k=K_1}^{K_2} \alpha_k \geq 0.9</math>, the coloring of the seven vertices <math>{\bf c}(0), {\bf c}(P e^{\pi ij/3}, j=1,\dots,6</math> is (up to rotation and recoloring) one of the three patterns of the central and linking vertices in Figure 3 of Aubrey's paper, namely

* <math>{\bf c}(0) = {\bf c}(P) = {\bf c}(P e^{\pi i/3}) = {\bf c}(P e^{2\pi i/3}) = {\bf c}(P e^{3\pi i/3}) = {\bf c}(P e^{4\pi i/3}) = {\bf c}(P e^{5\pi i/3}) </math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{3\pi i/3})</math>; <math>{\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{3\pi i/3})</math>; <math> {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>.

Using the spindling argument from Aubrey's paper, we conclude that the third possibility must in fact hold with probability at least 0.8; on the other hand, from Lemma 2 this scenario can only occur with probability at most 1/2, giving the required contradiction.
<math>\Box</math>

One should be able to refine this argument to show that <math>p_{H = 1tri} > c</math> for an absolute constant <math> c>0 </math>.

=== Lemma 21 ===
Providing a tighter bound for Lemma 17 with a more thorough proof: If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths <math>a,b,c</math>. Let <math>\left|z_2\right|=b,\left|a-z_2\right|=c</math>. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also: <math>\mathbf{c}(a)\neq\mathbf{c}(z_2)\Rightarrow[\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)]</math>. <math>[A\Rightarrow B]\Rightarrow {\bf P}(A)\leq{\bf P}(B)</math> thus
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) \geq {\bf P}(\mathbf{c}(a) \neq \mathbf{c}(z_2)) = 1-p_c</math>
<math>{\bf P}(A\lor B) +{\bf P}(A\land B)={\bf P}(A)+{\bf P}(B)</math>, so
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)) + {\bf P}(\mathbf{c}(0)\neq\mathbf{c}(z_2)) - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>1-p_c \leq 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
Using the law of cosines: <math>z_2=b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)</math>
:<math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math> <math>\Box</math>

=== Lemma 22 ===

One has <math>3 p_{1/\sqrt{3}} \geq {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) )</math>.

'''Proof''' Let <math>z</math> be a complex number of magnitude <math>1/\sqrt{3}</math> that is a unit distance from 1. If <math>\mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) = c</math> (say), then <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> cannot be colored with <math>c</math>; also, <math>z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> are the vertices of a unit equilateral triangle and thus must take on three different colors. By the pigeonhole principle, one of <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> must then take the same color as the origin, and the claim follows. <math>\Box</math>

=== Lemma 23 ===

One has <math>4 p_{(\sqrt{6} \pm \sqrt{2})/2} + p_{\sqrt{2}} \geq 1</math>. In particular, by Lemma 2, <math>p_{(\sqrt{6}+\sqrt{2})/2} \geq 1/8</math>.

'''Proof''' [ExIs2018b] We just prove the claim for the + sign (the - sign can then be obtained after applying the Galois conjugacy that maps <math>\sqrt{3}</math> to <math>-\sqrt{3}</math>, leaving <math>\sqrt{2}</math> unchanged). Set <math>d := \frac{\sqrt{6}+\sqrt{2}}{2}</math>, and consider the five vertices

:<math>0, e^{5\pi i/4}, e^{5\pi i/4} + d, e^{5\pi i/4} + e^{\pi i/3} d, e^{5\pi i/4} + (e^{\pi i/3}-i)d.</math>

One can check that of the ten edges determined by these five vertices, five have unit length, four have length d, and the remaining distance (from 0 to <math>e^{5\pi i/4}+d</math>) has distance <math>\sqrt{2}</math>. Since <math>\mathbf{c}</math> must make one of the latter five edges monochromatic, the claim follows. <math>\Box</math>

=== Lemma 24 ===

One has <math>p_{\sqrt{2}} \geq \frac{1}{14}</math>.

'''Proof''' In page 7 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 20 unit distance edges and 14 edges of length <math>\sqrt{2}</math> is constructed. The coloring <math>\mathbf{c}</math> must make one of the 14 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 25 ===

Let <math>d = \frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math> and <math>e = \frac{3^{1/4} \sqrt{2} + \sqrt{3} - 1}{2}</math>.
Then one has <math>14 p_d + p_e \geq 1</math>. In particular, by Lemma 2, <math>p_d \geq 1/28</math>.

'''Proof''' In page 9 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 19 unit edges, 14 edges of length d, and one edge of length e is constructed. The coloring <math>\mathbf{c}</math> must make one of the 15 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 26 ===

Let <math>d = \sqrt{3/2 + \sqrt{33}/6}</math>. Then <math>7 p_d \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_d \geq \frac{1}{196}</math>.

'''Proof''' In page 11 of [ExIs2018b], a graph of nine vertices consisting of 12 unit edges and 7 edges of length d is constructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Thus, <math>\mathbf{c}</math> can only make the AB edge monochromatic if one of the seven length d edges is monochromatic. The claim follows.
<math>\Box</math>

=== Lemma 27 ===

One has <math>27 p_{\sqrt{5/3}} \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_{\sqrt{5/3}} \geq \frac{1}{756}</math>.

'''Proof''' In page 13 of [ExIs2018], a graph of 33 vertices with some unit edges and 27 edges of length <math>\sqrt{5/3}</math> is contructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Now repeat the proof of Lemma 26. <math>\Box</math>

=== Computer-verified claim 28 ===

One has <math>p_{2/\sqrt{3}} \geq \frac{1}{177}</math>.

'''Proof''' In page 15 of [ExIs2018], a 5-chromatic graph of 103 vertices, 312 unit edges, and 177 edges of length <math>2/\sqrt{3}</math> is constructed. <math>\mathbf{c}</math> must make one of the latter edges monochromatic, giving the claim. <math>\Box</math>

=== Lemma 29 ===

One has <math>p_{(\sqrt{6} \pm \sqrt{2})/2} \geq 1/6</math> (this improves the bound in Lemma 23).

'''Proof''' Use graphs 505 and 507 from [S2004] and the spindle bound. <math>\Box</math>

=== Lemma 30 ===
For <math>m > n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' <math>n</math> colors and <math>m</math> points necessitates at least 2 having equal color. I.e.

:<math>{\bf P}\left(\bigvee_{k=0}^n \bigvee_{j=k+1}^n\ \mathbf{c}(z_k) = \mathbf{c}(z_j)\right) = 1</math>

The lemma then follows immediately from the fact:
:<math>{\bf P}\left(\bigcup_{k} E_k\right) \leq \sum_{k} {\bf P}\left(E_k\right) \,\Box</math>

=== Corollary 31 ===
Let <math>\lvert z_k\rvert=1</math>. Then for <math>m \geq n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' Use lemma 30 on the set <math>\left\{z_k \bigg\vert 1\leq k\leq m \land k\in\mathbb{Z}\right\}\cup\{0\}</math>. Simplify using <math>{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(0) \right)=0</math>.
<math>\Box</math>

=== Lemma 32 ===
For <math>x\in\mathbb{R}</math> and an <math>n</math>-coloring of the plane, <math>\sum_{k=1}^{n-1}\left(n-k\right){\bf P}\left(\mathbf{c}\left(0\right) = \mathbf{c}\left( 2\sin\left(\frac{kx}{2}\right) \right) \right) \geq 1</math>.

'''Proof''' Use corollary 31 on the set <math>\left\{e^{ikx} \bigg\vert 0\leq k < n \land k\in\mathbb{Z}\right\}</math>. and simplify by grouping lengths.
<math>\Box</math>

=== Corollary 33 ===
Interesting(easy to simplify results of) values for <math>x</math> in Lemma 32 are in <math>\left\{x \bigg\vert \sin\left(\frac{kx}{2}\right)=1 \land 1\leq k < n \land k\in\mathbb{Z}\right\}</math>.
For 4-colorings, this gives
:<math>2p_{\sqrt 3}+p_2 \geq 1</math>
:<math>3p_{(\sqrt 3-1)/\sqrt 2}+p_{\sqrt 2} \geq 1</math>
:<math>3p_{2\sin(\pi/18)}+2p_{2\sin(\pi/9)} \geq 1</math>

=== Lemma 34 ===
Generalizing the note of Lemma 17, <math>\lvert d_1\rvert= d_1 > \lvert d_0\rvert= d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>

'''Proof''' let <math>\lvert z_{j+1} -z_j\rvert=d_0 > 0, \lvert z_{j+n} -z_0\rvert=d_1>0</math>.

Base case, <math>n=2</math>, by Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle:
:<math>2d_0\geq d_1\Rightarrow 2p_{d_0}\leq 1+p_{d_1}</math>
The inductive step is Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle. After induction:
:<math>[n\geq 2\land nd_0\geq d_1]\Rightarrow np_{d_0}\leq n-1+p_{d_1}</math>
Substitute <math>n=\left\lceil\frac{d_1}{d_0}\right\rceil</math>, simplify, rearrange:
:<math>d_1 > d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>
<math>\Box</math>

=== Proposition 35 ===

If <math>d > 1/\sqrt{2}</math> obeys the inequalities
:<math>\sin( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
and
:<math>\cos( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
then
:<math>p_d \leq \frac{1}{2} - \frac{1}{188}.</math>

(One can check that the conditions are obeyed precisely when <math>d \geq \frac{\sqrt{33}-1}{8} = 0.84307\dots</math>.)

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. Since <math>AB</math> is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the triangle <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>.

Now let <math>BADE</math> be a rhombus with sidelengths d and <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. By the hypotheses, the diagonals BD, AE of this rhombus have length at least 1/2, and hence are monochromatic with probability at most 1/2 by Lemma 2. As above, ABD and BDE are each monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>. As BD is monochromatic with probability at most <math>1/2</math>, we conclude that BADE is monochormatic with probability at least <math>\frac{1}{2}-6\delta</math>.

Now let <math>EDFG</math> be another rhombus congruent to <math>BADE</math>. As BD, AE have length at least 1/2, at least one of the long diagonals BF, AG have length at least 1/2 (the diagonal opposite an obtuse or right-angled triangle will work). Let's say BF has length at least 1/2. As BADE and EDFG are both monochromatic with probability at least <math>\frac{1}{2}-6\delta</math>, and the common edge DE is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the entire configuration ABDEFG is monochromatic with probability at least <math>\frac{1}{2}-11\delta</math>. In particular the pentagon ABDEF is monochromatic with at least this probability. However, in this pentagon, the five edges BA, AD, DE, EB, EF are monochromatic with probability at most <math>\frac{1}{2}-\delta</math>, and the other five edges are monochromatic with probability at most <math>\frac{1}{2}</math> by Lemma 2. Thus the probability that at least one of the edges of this pentagon is monochromatic is at most <math>(\frac{1}{2}-11\delta) + 5 \times 10\delta + 5 \times 11\delta = \frac{1}{2}+94\delta</math>. On the other hand, by the pigeonhole principle, this probability is 1. The claim follows. <math>\Box</math>

=== Proposition 36 ===

If <math>d \ge \frac{2}{\sqrt{15}} = 0.5163\dots</math>, then <math>p_d \leq \frac{1}{2} - \frac{1}{62}</math>,

if <math>d \ge \frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math>, then <math>p_d \leq 0.48</math>,

and <math>\limsup_d p_d\leq \frac{323}{675}=0.4785\ldots</math> (so <math>p_d<0.4786</math> if <math>d</math> is large enough).

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. A simple calculation shows that if <math>d \ge \frac{2}{\sqrt{15}}</math>, then <math>|BD| \ge \frac{1}{2}</math>.
If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. By inclusion-exclusion, we conclude that outside of the event that <math>AB</math> is monochromatic, the probability that <math>AD</math> is monochromatic is at most <math>2\delta</math>.

Now let <math>ADE</math> the <math>180^\circ</math> rotation of <math>ADB</math> around the midpoint of <math>AD</math>, and let <math>FDE</math> be the <math>180^\circ</math> rotation of <math>ADE</math> around the midpoint of <math>DE</math>. By the hypotheses, the line segments <math>AE, BD, BE, BF, DF</math> all have length at least 1/2. Let <math>{\mathcal E}</math> be the event that at least one of <math>AB, AD, DE, EF</math> is monochromatic. By the previous paragraph, this event occurs with probability at most <math>\frac{1}{2}-\delta+2\delta+2\delta+2\delta = \frac{1}{2}+5\delta</math>.

By previous considerations, <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>, and this event lies in <math>{\mathcal E}</math>. On the other hand, <math>BD</math> is monochromatic with probability at most 1/2 by Lemma 2. We conclude that outside of <math>{\mathcal E}</math>, <math>BD</math> is only monochromatic with probability at most <math>3\delta</math>. A similar argument (replacing <math>ABD</math> by <math>DAE</math> or <math>EDF</math>) shows that outside of <math>{\mathcal E}</math>, <math>AE</math> is monochromatic with probability at most <math>3\delta</math>, and similarly for <math>DF</math>. Now we consider <math>EB</math>. By previous considerations, the probability that <math>ABDE</math> is monochromatic is at least <math>\frac{1}{2}-5\delta</math>, and this event lies inside <math>{\mathcal E}</math>. Thus, outside of <math>{\mathcal E}</math>, the probability that <math>EB</math> is monochromatic is at most <math>5\delta</math>; similarly for <math>AF</math>. Finally, the probability that <math>BF</math> is monochromatic outside of <math>{\mathcal E}</math> is at most <math>7\delta</math>. We conclude that outside of an event of probability
:<math>\frac{1}{2}+5\delta + 3\delta+3\delta+3\delta+5\delta+5\delta+7\delta = \frac{1}{2} + 31\delta,</math>
none of the ten edges connecting <math>A,B,D,E,F</math> are monochromatic. But by the pigeonhole principle, this cannot occur in a 4-coloring, hence <math>\frac{1}{2} + 31 \delta \geq 1</math>, and the first claim follows.

For the second claim, we need to use an iterative argument, by feeding the bounds obtained back into the place in the proof where Lemma 2 is currently invoked. To have all occurring distances stay larger than <math>d</math>, we only need to check <math>|BD| \ge d</math>. Equality occurs when <math>ABD</math> is an equilateral triangle, which means that <math>ABC</math> and <math>ACD</math> are isosceles triangles with sides <math>d,d,1</math> and either with angles <math>150^\circ,15^\circ,15^\circ</math>, or with angles <math>30^\circ,75^\circ,75^\circ</math>. From here calculation gives <math>d \ge \frac{1}{2sin(75^\circ)}=\frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math> and <math>d \le \frac{1}{2sin(15^\circ)}=\frac{\sqrt{3}+1}{\sqrt{2}}=1.9318\dots</math>, but the upper bound is not really important, as for us it is enough that <math>|BD|</math> always stay above <math>d_0=\frac{\sqrt{3}-1}{\sqrt{2}}</math>, which occurs everywhere above this value. Now pick a <math>d\ge d_0</math> for which <math>p_d\ge \frac{1}{2}-\delta-\varepsilon</math>, where <math>\sup_{d\ge d_0} p_d= \frac{1}{2}-\delta</math> and <math>\varepsilon</math> is a small positive number. The calculation of the first case gives <math>\frac{1}{2}+5\delta + 2\delta+2\delta+2\delta+4\delta+4\delta+6\delta+O(\varepsilon) =\frac{1}{2} + 25 \delta+O(\varepsilon) \geq 1</math>, from which <math>\delta\ge 0.02</math> if we choose <math>\varepsilon</math> small enough.

To prove the last claim, we modify the construction; we obtain <math>F</math> by reflecting <math>B</math> to <math>AD</math>, to win <math>2\delta</math> in the last step of the calculation. To invoke Lemma 2, we need (among other things) that <math>EF</math> is at least 1/2, and to iterate in a straight-forward way, we would need a value <math>d_0</math> for which <math>EF</math> is at least <math>d_0</math> if <math>AB</math> is <math>d\ge d_0</math>, but such a <math>d_0</math> doesn't exist. We can, however, still iterate in a weaker sense, as <math>6</math> of the occurring <math>10</math> distances tend to infinity as <math>d=|AB|</math> tends to infinity, and the remaining <math>4</math> are also larger than <math>\frac{\sqrt{3}-1}{\sqrt{2}}</math>, so their probability of them being monochromatic is at most <math>0.48=(0.5-\delta)+(\delta-0.02)</math>. What we get eventually is <math>\frac{1}{2} + 25 \delta-2\delta+ 4(\delta-0.02)+O(\varepsilon) =0.42 + 27 \delta+O(\varepsilon) \geq 1</math>, from which <math>p_d\le \frac{323}{675}=0.4785\ldots</math> for <math>d</math> large enough.
<math>\Box</math>

=== Lemma 37 ===

One has <math>\sup_{0 < d < 2} p_d \geq 1/3</math>.

'''Proof''' For a large integer <math>n</math>, consider the points <math>e^{2\pi i j/n}</math> with <math>j=1,\dots,n</math>. Any unit distance coloring will color these points in at most 3 colors, hence divides the n points into three color classes of some size <math>n_1,n_2,n_3</math>. The number of monochromatic pairs is then
:<math>\frac{n_1(n_1-1)}{2} + \frac{n_2(n_2-1)}{2} + \frac{n_3(n_3-1)}{2} = \frac{1}{2} (n_1^2+n_2^2+n_3^2) + O(n) \geq \frac{1}{6} n^2 + O(n)</math>
by Cauchy-Schwarz. Thus at least <math>1/3-O(1/n)</math> of the pairs are monochromatic. Taking expectations and using the pigeonhole principle, we conclude that one of the distances has a probability at least <math>1/3 -O(1/n)</math> of being monochromatic, giving the claim. <math>\Box</math>

=== Lemma 38 ===

Let ABC be a unit-edge equilateral triangle, and let D be an arbitrary point. Let <math>|AD|, |BD|</math> and <math>|CD|</math> be <math>d,e,f</math> respectively. Then <math>p(d)+p(e)+p(f) \leq 1</math>. In particular, examining the case <math>e=f</math>, if <math>p(d) \geq k</math> then <math>p(\sqrt(d(d \pm \sqrt 3) + 1) \leq (1-k)/2</math>.

'''Proof''' At most one of <math>AD, BD</math> and <math>CD</math> can be monochromatic. <math>\Box</math>

Note: A consequence is that a 4-chromatic unit-distance graph G can demonstrate CNP <math>> 4</math> if, for the {d,e,f} arising from some choice of D above, G contains three equal-sized non-empty sets v_d, v_e, v_f of vertex-pairs such that (a) each vertex-pair within v_d is at distance d (resp. e and f), and (b) in any 4-colouring of G, more than 1/3 of the vertex-pairs in the union of the three sets are monochromatic. Note that this demonstration does not require that v_d contain all the vertex-pairs of G that are at distance d (resp. e and f), nor even that the graph {A,B,C,D} which gives rise to {d,e,f} be a subgraph of G. It seems plausible to find such a graph that is small (and/or symmetrical) enough that its colourings can be human-analysed to establish this property.

== Simplification rules for triplets of points in the complex plane ==
Deduced from the rule <math>{\bf P}(A\land B)+{\bf P}(A\land \lnot B)={\bf P}(A)</math>.
:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) = {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) - {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) ) - {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) \neq {\mathbf c}(z_0) ) + {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) = {\mathbf c}(z_0) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

== Proofs of bounds for conditional probabilities ==
The trivial case, valid where <math>\left|d\right|\neq 1</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) = {\mathbf c}(d) )=0</math>

Trivial plus Baye's Theorem, valid where <math>d\neq 0</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) )=\frac{{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )}\leq 1</math>
Rearrange:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )+{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1</math>

Spindle method: for <math>\left|d\right|=d\geq 1/2</math> and <math>\theta=2\text{arcsin}\left(\frac{1}{2d}\right)</math>:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{i\theta}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) ) = \frac{1}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )} - 1\leq 1</math>
which is another way to see <math>\left|d\right|=d\geq 1/2\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1/2</math>.

== Further questions ==
* For <math>n,m\geq CNP</math>, what consistent relationships exist between <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert n\text{ colors}\right)</math> and <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert m\text{ colors}\right)</math>? How can these relationships be used to sharpen arguments of the probabilistic formulation?

[[Category:Polymath16]]

Probabilistic formulation of Hadwiger-Nelson problem

2018-07-18T10:42:43Z

Aubrey: /* Bounds on p_d for 4-colourings */

Suppose for sake of contradiction that we have a 4-coloring <math>c: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with no unit edges monochromatic, thus

:<math>c(z) \neq c(w) \hbox{ whenever } |z-w| = 1. \quad (1)</math>

We can create further such colorings by composing <math>c</math> on the left with a permutation <math>\sigma \in S_4</math> on the left, and with the (inverse of) a Euclidean isometry <math>T \in E(2)</math> on the right, thus creating a new coloring <math>\sigma \circ c \circ T^{-1}: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with the same property. This is an action of the solvable group <math>S_4 \times E(2)</math>.

It is a fact that all solvable groups (viewed as discrete groups) are [https://en.wikipedia.org/wiki/Amenable_group amenable], so in particular <math>S_4 \times E(2)</math> is amenable. This means that there is a finitely additive probability measure <math>\mu</math> on <math>S_4 \times E(2)</math> (with all subsets of this group measurable), which is left-invariant: <math>\mu(gE) = \mu(E)</math> for all <math>g \in S_4 \times E(2)</math> and <math>E \subset S_4 \times E(2)</math>. This gives <math>S_4 \times E(2)</math> the structure of a finitely additive probability space. We can then define a random coloring <math>{\mathbf c}: {\bf C} \to \{1,2,3,4\}</math> by defining <math>{\mathbf c} := {\mathbf \sigma} \circ c \circ {\mathbf T}^{-1}</math>, where <math>({\mathbf \sigma},{\mathbf T})</math> is the element of the sample space <math>S_4 \times E(2)</math>. Thus for any complex number <math>z</math>, the random color <math>{\mathbf c}(z)</math> is a random variable taking values in <math>\{1,2,3,4\}</math>. The left-invariance of the measure implies that for any <math>(\sigma,T) \in S_4 \times E(2)</math>, the coloring <math> \sigma \circ {\mathbf c} \circ T^{-1}</math> has the same law as <math>{\mathbf c}</math>. This gives the color permutation invariance

:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(z_1) = \sigma(c_1), \dots, {\mathbf c}(z_k) = \sigma(c_k) )\quad (2)</math>

for any <math>z_1,\dots,z_k \in {\bf C}</math>, <math>c_1,\dots,c_k \in \{1,2,3,4\}</math>, and <math>\sigma \in S_4</math>, and the Euclidean isometry invariance
:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(T(z_1)) = c_1, \dots, {\mathbf c}(T(z_k)) = c_k. \quad (3)</math>
(In probabilistic language, this means that the random coloring is a [https://en.wikipedia.org/wiki/Stationary_process stationary process] with respect to the action of <math>S_4 \times E(2)</math>. The extraction of a stationary process from a deterministic object is an example of the ''Furstenberg correspondence principle''.)

== Bounds on <math>p_d</math> for 4-colourings ==

A class of correlations that is of particular interest is that of vertex pairs at some distance <math>d</math>. Accordingly, define
:<math>p_d := {\bf P}( \mathbf{c}(0) = \mathbf{c}(d) ).</math>

{| border=1
|-
! distance !! Lower bound !! Lower-bounding graph/method !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math> \geq 1/2</math>
|
|
| 1/2
| Spindle
|
|-
| <math>\geq \frac{2}{\sqrt{15}}</math>
|
|
| <math>\frac{1}{2} - \frac{1}{62}</math>
| Proposition 36
|
|-
| <math>\geq \frac{\sqrt{3}-1}{\sqrt{2}}</math>
|
|
| <math>0.48</math>
| Proposition 36
|
|-
| large enough
|
|
| <math>\frac{323}{675}=0.4785\ldots</math>
| Proposition 36
|
|-
| <math> d\geq 1/n, n>1</math>
|
|
| <math>1-\frac{1}{n}</math>
| (n+1)-gon with one edge length 1 and the rest d, Lemma 34
|
|-
| <math> d\geq 1/(n \sqrt{3}), n>1</math>
|
|
| <math>(3n-2)/3n</math>
| (n+1)-gon with one edge length <math>1/\sqrt{3}</math> and the rest d, Lemma 34
| Not better than the above on intervals <math>\left(\frac{1}{7},\frac{1}{4\sqrt{3}}\right),\left(\frac{1}{4},\frac{1}{2\sqrt{3}}\right)</math>
|-
| 1
| 0
| Unit edge
| 0
| Unit edge
|
|-
| <math>\frac{1}{\sqrt{3}}</math>
| 1/28
| Unit diamond plus centres of triangles, together with H, Corollary 16
| 1/3
| Unit triangle plus its centre
|
|-
| 2
| 1/4
| <math>G_{21}</math>
| <math>0.48</math>
| Proposition 36
| Lower bound computer verified
|-
| <math>{\sqrt{3}}</math>
| 1/4
| H, Corollary 16
| <math>0.48</math>
| Proposition 36
|
|-
| <math>{\sqrt{7}}</math>
|
|
| <math>3/8</math>
| Lemma 38 and Corollary 16
|
|-
| <math>{\frac{\sqrt{5}+1}{2}}</math>
| 1/5
| regular pentagon with unit side length
| 2/5
| regular pentagon with unit side length
|
|-
| <math>{\frac{\sqrt{5}-1}{2}}</math>
| 1/5
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
| 2/5
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
|
|-
| <math>{\sqrt{11/3}}</math>
| 1/118
| <math>G_{40}</math>
| <math>0.48</math>
| Proposition 36
| computer-verified
|-
| 8/3
| 1
| <math>V \oplus V \oplus H</math>
| <math>0.48</math>
| Proposition 36
| computer-verified; leads to contradiction
|-
| <math>\frac{\sqrt{6} \pm \sqrt{2}}{2}</math>
| 1/6
| An arrangement of five vertices
| <math>0.5</math> and <math>0.48</math>
| Proposition 36
|
|-
| <math>\sqrt{2}</math>
| 1/14
| A graph of 13 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>\frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math>
| 1/28
| A graph of 13 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>\sqrt{3/2 + \sqrt{33}/6}</math>
| 1/196
| A graph of 9 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>\sqrt{5/3}</math>
| 1/756
| A graph of 33 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>2/\sqrt{3}</math>
| 1/177
| A graph of 103 vertices
| <math>0.48</math>
| Proposition 36
| Computer-verified
|-
| <math>\frac{\sqrt{33} \pm 1}{2\sqrt{3}}</math>
| 1/2
| <math>G_{420}</math>
| <math>0.48</math>
| Proposition 36
| Computer-verified
|}

== Bounds on <math>{\bf P}( \mathbf{c}(0) = \mathbf{c}(d_1) \mid \mathbf{c}(0) \neq \mathbf{c}(d_0) )</math> for 4-colourings ==

{| border=1
|-
! <math>d_1</math> !! <math>d_0</math> !! Lower bound !! Lower-bounding graph !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math>1+(-1)^{1/3}</math>
| <math>2</math>
| 1/2
| H
|
|
| Equals <math>p_{\sqrt 3}/(1-p_2)</math>
|-
| <math>1+(-1)^{1/3}</math>
| <math>1+(-1)^{-1/3}</math>
|
|
| 1/2
| H
|
|}

== Proofs of bounds ==

One can compute some correlations of the coloring exactly:

=== Lemma 1 ===
Let <math>z,w \in {\bf C}</math> with <math>|z-w|=1</math>. Then
:<math> {\bf P}( \mathbf{c}(z) = c ) = \frac{1}{4}\quad (4)</math>
for all <math>c=1,\dots,4</math>,
:<math> {\bf P}( \mathbf{c}(z) = \mathbf{c}(w) ) = 0\quad (5),</math>
and
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c' ) = \frac{1}{12} \quad (6)</math>
for any distinct <math>c,c' \in \{1,2,3,4\}</math>. If <math>u</math> is at a unit distance from both z and w, then
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c'; \mathbf{c}(u) = c'' ) = \frac{1}{24} \quad (6')</math>

'''Proof''' By color invariance (2), the four probabilities in (4) are equal and sum to 1, giving (4). The claim (5) is immediate from (1). From (5) and color invariance, the 12 probabilities in (6) are equal and sum to 1, giving (6). The same argument gives (6').<math>\Box</math>

=== Lemma 2 ===
(Spindle argument) Let <math>|d| \geq 1/2</math>. Then
:<math> p_d \leq \frac{1}{2} \quad (7).</math>

'''Proof''' We can find an angle <math>\theta</math> with <math>|de^{i\theta}-d|=1</math>, then <math>\mathbf{c}(de^{i\theta}) \neq \mathbf{c}(d)</math> almost surely. This means that at least one of the events <math>\mathbf{c}(0) = \mathbf{c}(d)</math>, <math>\mathbf{c}(0) = \mathbf{c}(d e^{i\theta})</math> occurs with probability at most 1/2. The claim now follows from isometry invariance (3). <math>\Box</math>

=== Lemma 3 ===
(Using the K graph) We have
:<math>52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) + {\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} ) \geq 1 \quad (8).</math>

'''Proof''' Consider the 61-vertex graph <math>K</math> from [https://arxiv.org/abs/1804.02385 de Grey's paper]. It has 26 (isometric) copies of H, and thus 52 copies of the triangle <math>(1, e^{2\pi i/3}, e^{4\pi i/3})</math>. With probability at least <math>1 - 52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) </math>, none of these triangles are monochromatic. By the argument in that paper, this implies that the three linking diagonals <math>(-2, +2)</math>, <math>(-2 e^{2\pi i/3}, 2e^{2\pi i/3})</math>, <math>(-2 e^{4\pi i/3}, e^{-4\pi i/3})</math> are monochromatic. This gives the claim. <math>\Box</math>

=== Corollary 4 ===
(Existence of monochromatic <math>\sqrt{3}</math>-triangles) We have <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) \geq \frac{1}{104}</math>.

'''Proof''' The probability <math>{\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} )</math> is at most <math>{\bf P}( \mathbf{c}(-2) = \mathbf{c}(2)) = p_4</math>, which by Lemma 2 is at most 1/2. The claim now follows from Lemma 3. <math>\Box</math>

=== Computer-verified Claim 5 ===
(Using the graph M) One has <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = 0</math> (Note this contradicts Corollary 4).

'''Proof''' This simply reflects the fact that there is no 4-coloring of the 1345-vertex graph M from [https://arxiv.org/abs/1804.02385 de Grey's paper] with its central copy of H containing a monochromatic triangle. One can use other graphs for this purpose, such as the 278-vertex graph <math>M_1</math> or <math>V \oplus V \oplus H</math>. <math>\Box</math>

=== Computer-verified Claim 6 ===
(Using the graph <math>V \oplus V \oplus H</math>) One has <math>p_{8/3} = 1</math> (note this contradicts Lemma 2).

'''Proof''' This reflects the fact that every 4-coloring of <math>V \oplus V \oplus H</math> must assign the same color to 0 and 8/3. There is also a 745-vertex subgraph of <math>V \oplus V \oplus H</math> with the same property. <math>\Box</math>

=== Lemma 7 ===
(Using <math>G_{40}</math>) We have
:<math>59 p_{\sqrt{11/3}} + p_{8/3} \geq 1</math>.

'''Proof''' This reflects the fact that every 4-coloring of the 40-vertex graph <math>G_{40}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which none of the 59 pairs of vertices at distance <math>\sqrt{11/3}</math> apart, will assign the same color to 0 and 8/3. (This is presumably human-verifiable.) <math>\Box</math>

=== Corollary 8 ===
One has
:<math> p_{\sqrt{11/3}} \geq \frac{1}{118}</math>.

'''Proof''' Combine Lemma 7 and Lemma 2. <math>\Box</math>

=== Lemma 9 ===
(Using <math>G_{49}</math>) One has
:<math>18 {\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq p_{\sqrt{11/3}} .</math>

'''Proof''' This reflects the fact that every 4-coloring of the 49-vertex graph <math>G_{49}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which 0 and <math>\sqrt{11/3}</math> have the same color, at least one of the 18 copies of <math>(1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3)</math> is monochromatic. This is potentially human-verifiable. <math>\Box</math>

=== Corollary 10 ===
(Existence of monochromatic <math>1/\sqrt{3}</math>-triangles) One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq \frac{1}{2124}.</math>

'''Proof''' Combine Corollary 8 and Lemma 9. <math>\Box</math>

=== Computer-verified claim 11 ===
One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) = 0</math>. (This contradicts Corollary 10).

'''Proof''' This reflects the fact that the 627-vertex graph <math>G_{627}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] does not have any 4-colorings with <math>1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3</math> monochromatic. <math>\Box</math>

=== Lemma 12 ===
For certain special distances d, one can improve the bound in Lemma 2:

If <math>k \geq 1</math> is a natural number, <math>j\in\mathbb{Z}</math>, <math>\gcd(j,2k+1)=1</math>, and <math>r = \frac{1}{2} \csc\left(\frac{j\pi}{2k+1}\right)</math> then
:<math> p_r \leq \frac{k}{2k+1},</math>
thus for instance
:<math> p_{\frac{1}{\sqrt{3}}} \leq \frac{1}{3}.</math>

'''Proof''' Observe that the regular 2k+1-polygon <math>r, re^{2\pi i/(2k+1)}, r e^{4\pi i/(2k+1)}, \dots, r^{4k\pi i/(k+1)}</math> has unit side lengths. By the pigeonhole principle, we conclude that at most k of these vertices can have the same color as the origin, and the claim follows. <math>\Box</math>

On the other hand, for <math>k=2,j=1</math> we also know from the regular pentagon of unit sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}+1}{2}} \leq \frac{2}{5} \quad (9)</math>
since any 4-coloring of that pentagon has either one or two monochromatic diagonals.

Similarly, for <math>k=2,j=2</math> we also know from the regular pentagon of <math>\frac{\sqrt{5}-1}{2}</math> sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}-1}{2}} \leq \frac{2}{5}</math>
since any 4-coloring of that pentagon has either one or two monochromatic edges. More generally, if <math>a,b,c,d,e</math> are the diagonal lengths of a pentagon with unit sides, then
:<math> 1 \leq p_a + p_b + p_c + p_d + p_e \leq 2.</math>

=== Lemma 13 ===
We have
:<math> 7 p_{\frac{1}{\sqrt{3}}} \geq p_{\sqrt{3}}</math>.

'''Proof''' Consider the unit rhombus <math>0, 1, e^{i\pi/3}, e^{-i\pi/3}</math> together with the centers <math>e^{i\pi/6}/\sqrt{3}, e^{-i\pi/6}/\sqrt{3}</math>. With probability <math>p_{\sqrt{3}}</math>, the two far vertices <math>e^{i\pi/3}, e^{-i\pi/3}</math> are the same color, and then 0,1 will be two other colors. This forces either one of the centers <math>e^{i\pi/6}/\sqrt{3}</math> of a triangle to have a common color with one of the vertices of that triangle, or the two centers must have the same color. Thus in any event one of the seven edges of distance <math>1/\sqrt{3}</math> is monochromatic, giving the claim. <math>\Box</math>

=== Corollary 14 ===
We have <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{728}</math>.

This slightly improves upon the lower bound of 1/2124 coming from Corollary 10.

'''Proof''' Combine Corollary 4 and Lemma 13. <math>\Box</math>

=== Lemma 15 ===
One has
:<math>p_{\sqrt{3}} + p_2 \geq \frac{2}{3}</math> and <math>2 p_{\sqrt{3}} + p_2 \geq 1</math>.

'''Proof''' As noted in de Grey's paper, there are essentially four 4-colorings of H. H has six edges of length <math>\sqrt{3}</math> and three of length <math>2</math>. If we let a denote the number of monochromatic <math>\sqrt{3}</math> edges and b the number of monochromatic <math>2</math>-edges, we see from inspection of all four colorings that <math>(a,b)</math> is either <math>(6, 0), (4,0), (2, 1)</math>, or <math>(0,3)</math>. In particular, one always has <math>\frac{a}{6} + \frac{b}{3} \geq \frac{2}{3}</math> and <math>2\frac{a}{6} + \frac{b}{3} \geq 1</math>. Taking expectations, we obtain the claim. <math>\Box</math>

=== Corollary 16 ===
We have <math>p_2 \geq \frac{1}{6}</math>, <math>p_{\sqrt{3}} \geq \frac{1}{4} </math> and <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{28}</math>.

'''Proof''' Combine Lemma 2, Lemma 15, and Lemma 13. <math>\Box</math>

=== Lemma 17 ===
If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths a,b,c. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also, thus
:<math>{\bf P}(\mathbf{c}(0) \neq \mathbf{c}(a)) + {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(b)) \geq {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(c))</math>
and the claim follows. <math>\Box</math>

Note that Lemma 2 follows from the a=b, c=1 case of this lemma. Iterating this lemma starting with Lemma 2 we can also obtain slightly nontrivial upper bounds on <math>p_a</math> for small values of a, e.g. <math>p_a \leq 1 - 2^{-k}</math> when <math>a \geq 2^{-k}, k\in\mathbb{Z}^+</math>.

Further, we can generalise the a=b case to one in which the triangle is replaced by a (k+1)-gon of which one edge is 1 and the others are all equal, leading to the stronger result <math>p_a \leq 1 - 1/k</math> when <math>a \geq 1/k, k\in\mathbb{Z}^+ \land k>1</math>. Further strengthening is achieved by using <math>1/\sqrt{3}</math> as the long edge, given Lemma 12.

=== Lemma 18 ===
Whenever <math>d>0</math>, one has the inequalities
:<math> |p_{\phi d} - p_d| \leq \frac{2}{5}, p_{\phi d} + p_d \geq \frac{1}{5}, 2p_d - p_{\phi d} \leq 1, 2 p_{\phi d} - p_d \leq 1</math>
where <math>\phi := \frac{1+\sqrt{5}}{2}</math> is the golden ratio. Also we have
:<math> p_{d/\sqrt{3}} \leq \frac{1}{3} + p_d, \frac{1}{2} + \frac{1}{2} p_d.</math>

Note that this generalises (9), as well as a special case of Lemma 12.

'''Proof''' Consider the regular pentagon with sidelength <math>d</math>, so it also has 5 diagonals of length <math>\phi d</math>. Let <math>a \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic edges and let <math>b \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic diagonals. Observe:
* <math>a,b</math> cannot both be zero (pigeonhole principle).
* <math>a</math> cannot be 4. Similarly, <math>b</math> cannot be 4.
* If <math>a=5</math>, then <math>b=5</math>, and conversely.
* If <math>a=0</math>, then <math>b=1,2</math>; similarly, if <math>b=0</math>, then <math>a=1,2</math>.

From this we observe the inequalities
:<math> |\frac{a}{5}-\frac{b}{5}| \leq \frac{2}{5}; \frac{a}{5} + \frac{b}{5} \geq \frac{1}{5}; 2 \frac{a}{5} - \frac{b}{5} \leq 1; 2\frac{b}{5} - \frac{a}{5} \leq 1</math>
and on taking expectations we obtain the first claim. Similarly, if one considers the colorings of an equilateral triangle of sidelength <math>d</math> together with its center, and counts the numbers <math>a,b \in \{0,1,2,3\}</math> of monochromatic edges of length <math>d</math> and <math>d/\sqrt{3}</math> respectively, one observes that one always has <math>\frac{b}{3} \leq \frac{1}{3} + \frac{2}{3} \frac{a}{3}, \frac{1}{2} + \frac{1}{2} \frac{a}{3}</math>, and on taking expectations one obtains the claim.<math>\Box</math>

The hexagon <math>H</math> has essentially four distinct colorings: the coloring <math>\hbox{2tri}</math> with two triangles, the coloring <math>\hbox{1tri}</math> with one triangle, the coloring <math>\hbox{axisym}</math> that is symmetric around an axis, and the coloring <math>\hbox{centralsym}</math> that is symmetric around the central point. This gives four probabilities <math>p_{H = 2tri}, p_{H = 1tri}, p_{H = axisym}, p_{H = centralsym}</math> that sum to 1. By counting the number of monochromatic edges of length <math>\sqrt{3}, 2</math> respectively, one also obtains the identities
:<math>p_{\sqrt{3}} = p_{H = 2tri} + \frac{2}{3} p_{H = 1tri} + \frac{1}{3} p_{H = axisym}; \quad p_2 = \frac{1}{3} p_{H=axisym} + p_{H=centralsym}</math>
which reproves Lemma 15. Also
:<math> {\bf P}( \mathbf{c}(0) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = p_{H = 2tri} + \frac{1}{2} p_{H=1tri}.</math>
Any 4-coloring of L contains at least one triangle within one of its 52 copies of H, thus
:<math> p_{H = 2tri} + \frac{1}{2} p_{H=1tri} \geq \frac{1}{52}</math>
which reproves Corollary 4.

=== Lemma 19 ===
(Hubai) One has <math>p_{H = 1tri} + p_{H = axisym} \geq \frac{1}{10}</math>.

'''Proof''' Consider five copies of H centred at 0,1,2,3,4. With probability at least <math>1 - 5( p_{H = 1tri} + p_{H = axisym} )</math>, none of these copies of H are colored 1tri or axisym, and so must be colored 2tri or centralsym. One can check then that if one of the copies is colored 2tri, then so is any adjacent copy; thus all five copies are colored 2tri, or all five are colored centralsym. In either case we see that -1 and 5 are colored the same color. Comparing with Lemma 2 then gives the claim. <math>\Box</math>

=== Theorem 20 ===
One has <math>p_{H = 1tri} > 0</math>.

'''Proof''' Suppose for contradiction that <math>p_{H = 1tri} = 0</math>. One can then run a version of the de Bruijn-Erdos argument to obtain a coloring in which 1tri hexagons are completely nonexistent (since there are arbitrarily large finite colorings with this property). Consider the triangular lattice <math>{\bf Z}[e^{\pi i/3}]</math>. We 2-color the edges of this lattice by coloring an edge black if it is the short diagonal of a unit rhombus with monochromatic long diagonal, and white otherwise. The four colorings of hexagons lead to four possible colorings at each vertex:

* If H is colored 2tri, then all six edges to the centre of H are black.
* If H is colored 1tri, then two edges to the centre of H at 120 degree angles are white, the other four are black.
* If H is colored axisym, then two opposing edges of the centre of H are black, the other four are white.
* If H is colored centralsym, then all six edges to the centre of H are black.

In particular, as we are assuming no 1tri hexagons, the faces cut out by the black edges have angles 60 degrees, and thus must be equilateral triangles, sectors of angle 60, half-planes, or the entire plane. If there is at least one equilateral triangle, then the rest of the black edges must form an equilateral lattice with that triangle sidelength. This leads to only a small number of possible hexagon colorings in the lattice:

# Case 1: All edges white.
# Case 2: All edges black.
# Case 3.k: For some natural number <math>k \geq 2</math>, the length k edges joining adjacent vertices in some coset of <math>k \cdot {\mathbf Z}[ e^{\pi i/3} ]</math> are all black, and the remaining edges are white.
# Case 4: Each horizontal row consists either entirely of black edges, or entirely of white edges.
# Case 5: Each northwest row consists either entirely of black edges, or entirely of white edges.
# Case 6: Each northeast row consists either entirely of black edges, or entirely of white edges.
# Case 7: Six rays of black edges meeting at a common vertex; all other edges white.

Technically, Case 1 is contained in Cases 4,5,6 as written above, but this will not be an issue. One can view Case 7 as a limiting case <math>k=\infty</math> of Case 3.k; Case 2 is similarly the opposite limiting case <math>k=1</math>.

In the first case, the coloring is periodic with periods <math>2, 2 e^{\pi i/3}</math>. In the second case, it is periodic with periods <math>3, 3 e^{\pi i/3}</math>. In the third case, it is periodic with periods <math>3k, 3k e^{\pi i/3}</math>. Also note that for each k, one can check if Case 3.k holds by inspecting the coloring at a finite number of vertices. Thus the event that Case 3.k holds is "measurable" in the sense that a meaningful probability can be assigned. (But Cases 1,2,4,5,6 are not measurable events, they require an infinite number of points to be inspected, and the probability measure we are using is only finitely additive rather than infinitely additive.) In Case 4, the coloring is periodic with period 2; also, every coset of <math>2 \cdot {\bf Z}[e^{\pi i/3}]</math> is 2-colored. Similarly for Case 5 and 6 (where the periods are <math>2 e^{2\pi i/3}</math> and <math>2 e^{4\pi i/3}</math> respectively.)

Let <math>\alpha_k</math> be the probability that Case 3.k holds for the given value of <math>k</math>. Then <math> \sum_{k=2}^K \alpha_k \leq 1</math> for any k, hence <math>\sum_{k=2}^\infty \alpha_k \leq 1</math>. In particular, we can find <math>K_1</math> such that <math>\sum_{k={K_1}}^\infty \alpha_k \leq 0.1</math> (say). Let <math>P</math> be six times the least common multiple of <math>1,2,\dots,K_1</math>. Then the coloring is P- and <math>P e^{\pi i/3}</math>-periodic for Case 1, Case 2, and all Case 3.k with <math>k \leq K_1</math>. On the other hand, if <math>K_2</math> is sufficiently large depending on <math>P</math>, and Case 3.k holds for some <math>k \geq K_2</math>, then almost all of the hexagons are colored centralsym, which makes the coloring "almost <math>P, P e^{\pi i/3}</math>-periodic" in the sense that
:<math>{\bf c}(z+P e^{\pi i j/3}) = {\bf c}(z) \hbox{ for } j=0,1,2,3,4,5</math>
will hold for at least <math>0.9</math> of the lattice points <math>z \in {\bf Z}[e^{\pi i/3}]</math> with <math>|z| \leq K_2</math>. Similarly for Case 7 (which is sort of a <math>k=\infty</math> limiting case of Case 3.k.) Thus, with the probability <math> \geq 1 - \sum_{k=K_1}^{K_2} \alpha_k \geq 0.9</math>, the coloring of the seven vertices <math>{\bf c}(0), {\bf c}(P e^{\pi ij/3}, j=1,\dots,6</math> is (up to rotation and recoloring) one of the three patterns of the central and linking vertices in Figure 3 of Aubrey's paper, namely

* <math>{\bf c}(0) = {\bf c}(P) = {\bf c}(P e^{\pi i/3}) = {\bf c}(P e^{2\pi i/3}) = {\bf c}(P e^{3\pi i/3}) = {\bf c}(P e^{4\pi i/3}) = {\bf c}(P e^{5\pi i/3}) </math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{3\pi i/3})</math>; <math>{\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{3\pi i/3})</math>; <math> {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>.

Using the spindling argument from Aubrey's paper, we conclude that the third possibility must in fact hold with probability at least 0.8; on the other hand, from Lemma 2 this scenario can only occur with probability at most 1/2, giving the required contradiction.
<math>\Box</math>

One should be able to refine this argument to show that <math>p_{H = 1tri} > c</math> for an absolute constant <math> c>0 </math>.

=== Lemma 21 ===
Providing a tighter bound for Lemma 17 with a more thorough proof: If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths <math>a,b,c</math>. Let <math>\left|z_2\right|=b,\left|a-z_2\right|=c</math>. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also: <math>\mathbf{c}(a)\neq\mathbf{c}(z_2)\Rightarrow[\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)]</math>. <math>[A\Rightarrow B]\Rightarrow {\bf P}(A)\leq{\bf P}(B)</math> thus
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) \geq {\bf P}(\mathbf{c}(a) \neq \mathbf{c}(z_2)) = 1-p_c</math>
<math>{\bf P}(A\lor B) +{\bf P}(A\land B)={\bf P}(A)+{\bf P}(B)</math>, so
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)) + {\bf P}(\mathbf{c}(0)\neq\mathbf{c}(z_2)) - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>1-p_c \leq 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
Using the law of cosines: <math>z_2=b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)</math>
:<math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math> <math>\Box</math>

=== Lemma 22 ===

One has <math>3 p_{1/\sqrt{3}} \geq {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) )</math>.

'''Proof''' Let <math>z</math> be a complex number of magnitude <math>1/\sqrt{3}</math> that is a unit distance from 1. If <math>\mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) = c</math> (say), then <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> cannot be colored with <math>c</math>; also, <math>z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> are the vertices of a unit equilateral triangle and thus must take on three different colors. By the pigeonhole principle, one of <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> must then take the same color as the origin, and the claim follows. <math>\Box</math>

=== Lemma 23 ===

One has <math>4 p_{(\sqrt{6} \pm \sqrt{2})/2} + p_{\sqrt{2}} \geq 1</math>. In particular, by Lemma 2, <math>p_{(\sqrt{6}+\sqrt{2})/2} \geq 1/8</math>.

'''Proof''' [ExIs2018b] We just prove the claim for the + sign (the - sign can then be obtained after applying the Galois conjugacy that maps <math>\sqrt{3}</math> to <math>-\sqrt{3}</math>, leaving <math>\sqrt{2}</math> unchanged). Set <math>d := \frac{\sqrt{6}+\sqrt{2}}{2}</math>, and consider the five vertices

:<math>0, e^{5\pi i/4}, e^{5\pi i/4} + d, e^{5\pi i/4} + e^{\pi i/3} d, e^{5\pi i/4} + (e^{\pi i/3}-i)d.</math>

One can check that of the ten edges determined by these five vertices, five have unit length, four have length d, and the remaining distance (from 0 to <math>e^{5\pi i/4}+d</math>) has distance <math>\sqrt{2}</math>. Since <math>\mathbf{c}</math> must make one of the latter five edges monochromatic, the claim follows. <math>\Box</math>

=== Lemma 24 ===

One has <math>p_{\sqrt{2}} \geq \frac{1}{14}</math>.

'''Proof''' In page 7 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 20 unit distance edges and 14 edges of length <math>\sqrt{2}</math> is constructed. The coloring <math>\mathbf{c}</math> must make one of the 14 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 25 ===

Let <math>d = \frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math> and <math>e = \frac{3^{1/4} \sqrt{2} + \sqrt{3} - 1}{2}</math>.
Then one has <math>14 p_d + p_e \geq 1</math>. In particular, by Lemma 2, <math>p_d \geq 1/28</math>.

'''Proof''' In page 9 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 19 unit edges, 14 edges of length d, and one edge of length e is constructed. The coloring <math>\mathbf{c}</math> must make one of the 15 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 26 ===

Let <math>d = \sqrt{3/2 + \sqrt{33}/6}</math>. Then <math>7 p_d \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_d \geq \frac{1}{196}</math>.

'''Proof''' In page 11 of [ExIs2018b], a graph of nine vertices consisting of 12 unit edges and 7 edges of length d is constructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Thus, <math>\mathbf{c}</math> can only make the AB edge monochromatic if one of the seven length d edges is monochromatic. The claim follows.
<math>\Box</math>

=== Lemma 27 ===

One has <math>27 p_{\sqrt{5/3}} \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_{\sqrt{5/3}} \geq \frac{1}{756}</math>.

'''Proof''' In page 13 of [ExIs2018], a graph of 33 vertices with some unit edges and 27 edges of length <math>\sqrt{5/3}</math> is contructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Now repeat the proof of Lemma 26. <math>\Box</math>

=== Computer-verified claim 28 ===

One has <math>p_{2/\sqrt{3}} \geq \frac{1}{177}</math>.

'''Proof''' In page 15 of [ExIs2018], a 5-chromatic graph of 103 vertices, 312 unit edges, and 177 edges of length <math>2/\sqrt{3}</math> is constructed. <math>\mathbf{c}</math> must make one of the latter edges monochromatic, giving the claim. <math>\Box</math>

=== Lemma 29 ===

One has <math>p_{(\sqrt{6} \pm \sqrt{2})/2} \geq 1/6</math> (this improves the bound in Lemma 23).

'''Proof''' Use graphs 505 and 507 from [S2004] and the spindle bound. <math>\Box</math>

=== Lemma 30 ===
For <math>m > n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' <math>n</math> colors and <math>m</math> points necessitates at least 2 having equal color. I.e.

:<math>{\bf P}\left(\bigvee_{k=0}^n \bigvee_{j=k+1}^n\ \mathbf{c}(z_k) = \mathbf{c}(z_j)\right) = 1</math>

The lemma then follows immediately from the fact:
:<math>{\bf P}\left(\bigcup_{k} E_k\right) \leq \sum_{k} {\bf P}\left(E_k\right) \,\Box</math>

=== Corollary 31 ===
Let <math>\lvert z_k\rvert=1</math>. Then for <math>m \geq n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' Use lemma 30 on the set <math>\left\{z_k \bigg\vert 1\leq k\leq m \land k\in\mathbb{Z}\right\}\cup\{0\}</math>. Simplify using <math>{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(0) \right)=0</math>.
<math>\Box</math>

=== Lemma 32 ===
For <math>x\in\mathbb{R}</math> and an <math>n</math>-coloring of the plane, <math>\sum_{k=1}^{n-1}\left(n-k\right){\bf P}\left(\mathbf{c}\left(0\right) = \mathbf{c}\left( 2\sin\left(\frac{kx}{2}\right) \right) \right) \geq 1</math>.

'''Proof''' Use corollary 31 on the set <math>\left\{e^{ikx} \bigg\vert 0\leq k < n \land k\in\mathbb{Z}\right\}</math>. and simplify by grouping lengths.
<math>\Box</math>

=== Corollary 33 ===
Interesting(easy to simplify results of) values for <math>x</math> in Lemma 32 are in <math>\left\{x \bigg\vert \sin\left(\frac{kx}{2}\right)=1 \land 1\leq k < n \land k\in\mathbb{Z}\right\}</math>.
For 4-colorings, this gives
:<math>2p_{\sqrt 3}+p_2 \geq 1</math>
:<math>3p_{(\sqrt 3-1)/\sqrt 2}+p_{\sqrt 2} \geq 1</math>
:<math>3p_{2\sin(\pi/18)}+2p_{2\sin(\pi/9)} \geq 1</math>

=== Lemma 34 ===
Generalizing the note of Lemma 17, <math>\lvert d_1\rvert= d_1 > \lvert d_0\rvert= d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>

'''Proof''' let <math>\lvert z_{j+1} -z_j\rvert=d_0 > 0, \lvert z_{j+n} -z_0\rvert=d_1>0</math>.

Base case, <math>n=2</math>, by Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle:
:<math>2d_0\geq d_1\Rightarrow 2p_{d_0}\leq 1+p_{d_1}</math>
The inductive step is Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle. After induction:
:<math>[n\geq 2\land nd_0\geq d_1]\Rightarrow np_{d_0}\leq n-1+p_{d_1}</math>
Substitute <math>n=\left\lceil\frac{d_1}{d_0}\right\rceil</math>, simplify, rearrange:
:<math>d_1 > d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>
<math>\Box</math>

=== Proposition 35 ===

If <math>d > 1/\sqrt{2}</math> obeys the inequalities
:<math>\sin( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
and
:<math>\cos( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
then
:<math>p_d \leq \frac{1}{2} - \frac{1}{188}.</math>

(One can check that the conditions are obeyed precisely when <math>d \geq \frac{\sqrt{33}-1}{8} = 0.84307\dots</math>.)

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. Since <math>AB</math> is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the triangle <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>.

Now let <math>BADE</math> be a rhombus with sidelengths d and <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. By the hypotheses, the diagonals BD, AE of this rhombus have length at least 1/2, and hence are monochromatic with probability at most 1/2 by Lemma 2. As above, ABD and BDE are each monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>. As BD is monochromatic with probability at most <math>1/2</math>, we conclude that BADE is monochormatic with probability at least <math>\frac{1}{2}-6\delta</math>.

Now let <math>EDFG</math> be another rhombus congruent to <math>BADE</math>. As BD, AE have length at least 1/2, at least one of the long diagonals BF, AG have length at least 1/2 (the diagonal opposite an obtuse or right-angled triangle will work). Let's say BF has length at least 1/2. As BADE and EDFG are both monochromatic with probability at least <math>\frac{1}{2}-6\delta</math>, and the common edge DE is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the entire configuration ABDEFG is monochromatic with probability at least <math>\frac{1}{2}-11\delta</math>. In particular the pentagon ABDEF is monochromatic with at least this probability. However, in this pentagon, the five edges BA, AD, DE, EB, EF are monochromatic with probability at most <math>\frac{1}{2}-\delta</math>, and the other five edges are monochromatic with probability at most <math>\frac{1}{2}</math> by Lemma 2. Thus the probability that at least one of the edges of this pentagon is monochromatic is at most <math>(\frac{1}{2}-11\delta) + 5 \times 10\delta + 5 \times 11\delta = \frac{1}{2}+94\delta</math>. On the other hand, by the pigeonhole principle, this probability is 1. The claim follows. <math>\Box</math>

=== Proposition 36 ===

If <math>d \ge \frac{2}{\sqrt{15}} = 0.5163\dots</math>, then <math>p_d \leq \frac{1}{2} - \frac{1}{62}</math>,

if <math>d \ge \frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math>, then <math>p_d \leq 0.48</math>,

and <math>\limsup_d p_d\leq \frac{323}{675}=0.4785\ldots</math> (so <math>p_d<0.4786</math> if <math>d</math> is large enough).

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. A simple calculation shows that if <math>d \ge \frac{2}{\sqrt{15}}</math>, then <math>|BD| \ge \frac{1}{2}</math>.
If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. By inclusion-exclusion, we conclude that outside of the event that <math>AB</math> is monochromatic, the probability that <math>AD</math> is monochromatic is at most <math>2\delta</math>.

Now let <math>ADE</math> the <math>180^\circ</math> rotation of <math>ADB</math> around the midpoint of <math>AD</math>, and let <math>FDE</math> be the <math>180^\circ</math> rotation of <math>ADE</math> around the midpoint of <math>DE</math>. By the hypotheses, the line segments <math>AE, BD, BE, BF, DF</math> all have length at least 1/2. Let <math>{\mathcal E}</math> be the event that at least one of <math>AB, AD, DE, EF</math> is monochromatic. By the previous paragraph, this event occurs with probability at most <math>\frac{1}{2}-\delta+2\delta+2\delta+2\delta = \frac{1}{2}+5\delta</math>.

By previous considerations, <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>, and this event lies in <math>{\mathcal E}</math>. On the other hand, <math>BD</math> is monochromatic with probability at most 1/2 by Lemma 2. We conclude that outside of <math>{\mathcal E}</math>, <math>BD</math> is only monochromatic with probability at most <math>3\delta</math>. A similar argument (replacing <math>ABD</math> by <math>DAE</math> or <math>EDF</math>) shows that outside of <math>{\mathcal E}</math>, <math>AE</math> is monochromatic with probability at most <math>3\delta</math>, and similarly for <math>DF</math>. Now we consider <math>EB</math>. By previous considerations, the probability that <math>ABDE</math> is monochromatic is at least <math>\frac{1}{2}-5\delta</math>, and this event lies inside <math>{\mathcal E}</math>. Thus, outside of <math>{\mathcal E}</math>, the probability that <math>EB</math> is monochromatic is at most <math>5\delta</math>; similarly for <math>AF</math>. Finally, the probability that <math>BF</math> is monochromatic outside of <math>{\mathcal E}</math> is at most <math>7\delta</math>. We conclude that outside of an event of probability
:<math>\frac{1}{2}+5\delta + 3\delta+3\delta+3\delta+5\delta+5\delta+7\delta = \frac{1}{2} + 31\delta,</math>
none of the ten edges connecting <math>A,B,D,E,F</math> are monochromatic. But by the pigeonhole principle, this cannot occur in a 4-coloring, hence <math>\frac{1}{2} + 31 \delta \geq 1</math>, and the first claim follows.

For the second claim, we need to use an iterative argument, by feeding the bounds obtained back into the place in the proof where Lemma 2 is currently invoked. To have all occurring distances stay larger than <math>d</math>, we only need to check <math>|BD| \ge d</math>. Equality occurs when <math>ABD</math> is an equilateral triangle, which means that <math>ABC</math> and <math>ACD</math> are isosceles triangles with sides <math>d,d,1</math> and either with angles <math>150^\circ,15^\circ,15^\circ</math>, or with angles <math>30^\circ,75^\circ,75^\circ</math>. From here calculation gives <math>d \ge \frac{1}{2sin(75^\circ)}=\frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math> and <math>d \le \frac{1}{2sin(15^\circ)}=\frac{\sqrt{3}+1}{\sqrt{2}}=1.9318\dots</math>, but the upper bound is not really important, as for us it is enough that <math>|BD|</math> always stay above <math>d_0=\frac{\sqrt{3}-1}{\sqrt{2}}</math>, which occurs everywhere above this value. Now pick a <math>d\ge d_0</math> for which <math>p_d\ge \frac{1}{2}-\delta-\varepsilon</math>, where <math>\sup_{d\ge d_0} p_d= \frac{1}{2}-\delta</math> and <math>\varepsilon</math> is a small positive number. The calculation of the first case gives <math>\frac{1}{2}+5\delta + 2\delta+2\delta+2\delta+4\delta+4\delta+6\delta+O(\varepsilon) =\frac{1}{2} + 25 \delta+O(\varepsilon) \geq 1</math>, from which <math>\delta\ge 0.02</math> if we choose <math>\varepsilon</math> small enough.

To prove the last claim, we modify the construction; we obtain <math>F</math> by reflecting <math>B</math> to <math>AD</math>, to win <math>2\delta</math> in the last step of the calculation. To invoke Lemma 2, we need (among other things) that <math>EF</math> is at least 1/2, and to iterate in a straight-forward way, we would need a value <math>d_0</math> for which <math>EF</math> is at least <math>d_0</math> if <math>AB</math> is <math>d\ge d_0</math>, but such a <math>d_0</math> doesn't exist. We can, however, still iterate in a weaker sense, as <math>6</math> of the occurring <math>10</math> distances tend to infinity as <math>d=|AB|</math> tends to infinity, and the remaining <math>4</math> are also larger than <math>\frac{\sqrt{3}-1}{\sqrt{2}}</math>, so their probability of them being monochromatic is at most <math>0.48=(0.5-\delta)+(\delta-0.02)</math>. What we get eventually is <math>\frac{1}{2} + 25 \delta-2\delta+ 4(\delta-0.02)+O(\varepsilon) =0.42 + 27 \delta+O(\varepsilon) \geq 1</math>, from which <math>p_d\le \frac{323}{675}=0.4785\ldots</math> for <math>d</math> large enough.
<math>\Box</math>

=== Lemma 37 ===

One has <math>\sup_{0 < d < 2} p_d \geq 1/3</math>.

'''Proof''' For a large integer <math>n</math>, consider the points <math>e^{2\pi i j/n}</math> with <math>j=1,\dots,n</math>. Any unit distance coloring will color these points in at most 3 colors, hence divides the n points into three color classes of some size <math>n_1,n_2,n_3</math>. The number of monochromatic pairs is then
:<math>\frac{n_1(n_1-1)}{2} + \frac{n_2(n_2-1)}{2} + \frac{n_3(n_3-1)}{2} = \frac{1}{2} (n_1^2+n_2^2+n_3^2) + O(n) \geq \frac{1}{6} n^2 + O(n)</math>
by Cauchy-Schwarz. Thus at least <math>1/3-O(1/n)</math> of the pairs are monochromatic. Taking expectations and using the pigeonhole principle, we conclude that one of the distances has a probability at least <math>1/3 -O(1/n)</math> of being monochromatic, giving the claim. <math>\Box</math>

=== Lemma 38 ===

Let ABC be a unit-edge equilateral triangle, and let D be an arbitrary point. Let <math>|AD|, |BD|</math> and <math>|CD|</math> be <math>x,y,z</math> respectively. Then <math>p(x)+p(y)+p(z) \leq 1</math>. In particular, examining the case e=f, if <math>p(d) \geq k</math> then <math>p(\sqrt(d(d \pm \sqrt 3) + 1) \leq (1-k)/2</math>.

'''Proof''' At most one of <math>AD, BD</math> and <math>CD</math> can be monochromatic. <math>\Box</math>

Note: A consequence is that a 4-chromatic unit-distance graph G can demonstrate CNP <math>> 4</math> if, for the {x,y,z} arising from some choice of D above, G contains three equal-sized non-empty sets v_x, v_y, v_z of vertex-pairs such that (a) each vertex-pair within v_x is at distance x (resp. y and z), and (b) in any 4-colouring of G, more than 1/3 of the vertex-pairs in the union of the three sets are monochromatic. Note that this demonstration does not require that v_x contain all the vertex-pairs of G that are at distance x (resp. y and z), nor even that the graph {A,B,C,D} which gives rise to {x,y,z} be a subgraph of G. It seems plausible to find such a graph that is small (and/or symmetrical) enough that its colourings can be human-analysed to establish this property.

== Simplification rules for triplets of points in the complex plane ==
Deduced from the rule <math>{\bf P}(A\land B)+{\bf P}(A\land \lnot B)={\bf P}(A)</math>.
:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) = {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) - {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) ) - {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) \neq {\mathbf c}(z_0) ) + {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) = {\mathbf c}(z_0) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

== Proofs of bounds for conditional probabilities ==
The trivial case, valid where <math>\left|d\right|\neq 1</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) = {\mathbf c}(d) )=0</math>

Trivial plus Baye's Theorem, valid where <math>d\neq 0</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) )=\frac{{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )}\leq 1</math>
Rearrange:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )+{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1</math>

Spindle method: for <math>\left|d\right|=d\geq 1/2</math> and <math>\theta=2\text{arcsin}\left(\frac{1}{2d}\right)</math>:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{i\theta}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) ) = \frac{1}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )} - 1\leq 1</math>
which is another way to see <math>\left|d\right|=d\geq 1/2\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1/2</math>.

== Further questions ==
* For <math>n,m\geq CNP</math>, what consistent relationships exist between <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert n\text{ colors}\right)</math> and <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert m\text{ colors}\right)</math>? How can these relationships be used to sharpen arguments of the probabilistic formulation?

[[Category:Polymath16]]

Probabilistic formulation of Hadwiger-Nelson problem

2018-07-18T10:41:32Z

Aubrey: /* Bounds on p_d for 4-colourings */

Suppose for sake of contradiction that we have a 4-coloring <math>c: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with no unit edges monochromatic, thus

:<math>c(z) \neq c(w) \hbox{ whenever } |z-w| = 1. \quad (1)</math>

We can create further such colorings by composing <math>c</math> on the left with a permutation <math>\sigma \in S_4</math> on the left, and with the (inverse of) a Euclidean isometry <math>T \in E(2)</math> on the right, thus creating a new coloring <math>\sigma \circ c \circ T^{-1}: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with the same property. This is an action of the solvable group <math>S_4 \times E(2)</math>.

It is a fact that all solvable groups (viewed as discrete groups) are [https://en.wikipedia.org/wiki/Amenable_group amenable], so in particular <math>S_4 \times E(2)</math> is amenable. This means that there is a finitely additive probability measure <math>\mu</math> on <math>S_4 \times E(2)</math> (with all subsets of this group measurable), which is left-invariant: <math>\mu(gE) = \mu(E)</math> for all <math>g \in S_4 \times E(2)</math> and <math>E \subset S_4 \times E(2)</math>. This gives <math>S_4 \times E(2)</math> the structure of a finitely additive probability space. We can then define a random coloring <math>{\mathbf c}: {\bf C} \to \{1,2,3,4\}</math> by defining <math>{\mathbf c} := {\mathbf \sigma} \circ c \circ {\mathbf T}^{-1}</math>, where <math>({\mathbf \sigma},{\mathbf T})</math> is the element of the sample space <math>S_4 \times E(2)</math>. Thus for any complex number <math>z</math>, the random color <math>{\mathbf c}(z)</math> is a random variable taking values in <math>\{1,2,3,4\}</math>. The left-invariance of the measure implies that for any <math>(\sigma,T) \in S_4 \times E(2)</math>, the coloring <math> \sigma \circ {\mathbf c} \circ T^{-1}</math> has the same law as <math>{\mathbf c}</math>. This gives the color permutation invariance

:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(z_1) = \sigma(c_1), \dots, {\mathbf c}(z_k) = \sigma(c_k) )\quad (2)</math>

for any <math>z_1,\dots,z_k \in {\bf C}</math>, <math>c_1,\dots,c_k \in \{1,2,3,4\}</math>, and <math>\sigma \in S_4</math>, and the Euclidean isometry invariance
:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(T(z_1)) = c_1, \dots, {\mathbf c}(T(z_k)) = c_k. \quad (3)</math>
(In probabilistic language, this means that the random coloring is a [https://en.wikipedia.org/wiki/Stationary_process stationary process] with respect to the action of <math>S_4 \times E(2)</math>. The extraction of a stationary process from a deterministic object is an example of the ''Furstenberg correspondence principle''.)

== Bounds on <math>p_d</math> for 4-colourings ==

A class of correlations that is of particular interest is that of vertex pairs at some distance <math>d</math>. Accordingly, define
:<math>p_d := {\bf P}( \mathbf{c}(0) = \mathbf{c}(d) ).</math>

{| border=1
|-
! distance !! Lower bound !! Lower-bounding graph/method !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math> \geq 1/2</math>
|
|
| 1/2
| Spindle
|
|-
| <math>\geq \frac{2}{\sqrt{15}}</math>
|
|
| <math>\frac{1}{2} - \frac{1}{62}</math>
| Proposition 36
|
|-
| <math>\geq \frac{\sqrt{3}-1}{\sqrt{2}}</math>
|
|
| <math>0.48</math>
| Proposition 36
|
|-
| large enough
|
|
| <math>\frac{323}{675}=0.4785\ldots</math>
| Proposition 36
|
|-
| <math> d\geq 1/n, n>1</math>
|
|
| <math>1-\frac{1}{n}</math>
| (n+1)-gon with one edge length 1 and the rest d, Lemma 34
|
|-
| <math> d\geq 1/(n \sqrt{3}), n>1</math>
|
|
| <math>(3n-2)/3n</math>
| (n+1)-gon with one edge length <math>1/\sqrt{3}</math> and the rest d, Lemma 34
| Not better than the above on intervals <math>\left(\frac{1}{7},\frac{1}{4\sqrt{3}}\right),\left(\frac{1}{4},\frac{1}{2\sqrt{3}}\right)</math>
|-
| 1
| 0
| Unit edge
| 0
| Unit edge
|
|-
| <math>\frac{1}{\sqrt{3}}</math>
| 1/28
| Unit diamond plus centres of triangles, together with H, Corollary 16
| 1/3
| Unit triangle plus its centre
|
|-
| 2
| 1/4
| <math>G_{21}</math>
| <math>0.48</math>
| Proposition 36
| Lower bound computer verified
|-
| <math>{\sqrt{3}}</math>
| 1/4
| H, Corollary 16
| <math>0.48</math>
| Proposition 36
|
|-
| <math>{\sqrt{7}}</math>
|
|
| <math>3/8</math>
| Proposition 38 and Corollary 16
|
|-
| <math>{\frac{\sqrt{5}+1}{2}}</math>
| 1/5
| regular pentagon with unit side length
| 2/5
| regular pentagon with unit side length
|
|-
| <math>{\frac{\sqrt{5}-1}{2}}</math>
| 1/5
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
| 2/5
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
|
|-
| <math>{\sqrt{11/3}}</math>
| 1/118
| <math>G_{40}</math>
| <math>0.48</math>
| Proposition 36
| computer-verified
|-
| 8/3
| 1
| <math>V \oplus V \oplus H</math>
| <math>0.48</math>
| Proposition 36
| computer-verified; leads to contradiction
|-
| <math>\frac{\sqrt{6} \pm \sqrt{2}}{2}</math>
| 1/6
| An arrangement of five vertices
| <math>0.5</math> and <math>0.48</math>
| Proposition 36
|
|-
| <math>\sqrt{2}</math>
| 1/14
| A graph of 13 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>\frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math>
| 1/28
| A graph of 13 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>\sqrt{3/2 + \sqrt{33}/6}</math>
| 1/196
| A graph of 9 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>\sqrt{5/3}</math>
| 1/756
| A graph of 33 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>2/\sqrt{3}</math>
| 1/177
| A graph of 103 vertices
| <math>0.48</math>
| Proposition 36
| Computer-verified
|-
| <math>\frac{\sqrt{33} \pm 1}{2\sqrt{3}}</math>
| 1/2
| <math>G_{420}</math>
| <math>0.48</math>
| Proposition 36
| Computer-verified
|}

== Bounds on <math>{\bf P}( \mathbf{c}(0) = \mathbf{c}(d_1) \mid \mathbf{c}(0) \neq \mathbf{c}(d_0) )</math> for 4-colourings ==

{| border=1
|-
! <math>d_1</math> !! <math>d_0</math> !! Lower bound !! Lower-bounding graph !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math>1+(-1)^{1/3}</math>
| <math>2</math>
| 1/2
| H
|
|
| Equals <math>p_{\sqrt 3}/(1-p_2)</math>
|-
| <math>1+(-1)^{1/3}</math>
| <math>1+(-1)^{-1/3}</math>
|
|
| 1/2
| H
|
|}

== Proofs of bounds ==

One can compute some correlations of the coloring exactly:

=== Lemma 1 ===
Let <math>z,w \in {\bf C}</math> with <math>|z-w|=1</math>. Then
:<math> {\bf P}( \mathbf{c}(z) = c ) = \frac{1}{4}\quad (4)</math>
for all <math>c=1,\dots,4</math>,
:<math> {\bf P}( \mathbf{c}(z) = \mathbf{c}(w) ) = 0\quad (5),</math>
and
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c' ) = \frac{1}{12} \quad (6)</math>
for any distinct <math>c,c' \in \{1,2,3,4\}</math>. If <math>u</math> is at a unit distance from both z and w, then
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c'; \mathbf{c}(u) = c'' ) = \frac{1}{24} \quad (6')</math>

'''Proof''' By color invariance (2), the four probabilities in (4) are equal and sum to 1, giving (4). The claim (5) is immediate from (1). From (5) and color invariance, the 12 probabilities in (6) are equal and sum to 1, giving (6). The same argument gives (6').<math>\Box</math>

=== Lemma 2 ===
(Spindle argument) Let <math>|d| \geq 1/2</math>. Then
:<math> p_d \leq \frac{1}{2} \quad (7).</math>

'''Proof''' We can find an angle <math>\theta</math> with <math>|de^{i\theta}-d|=1</math>, then <math>\mathbf{c}(de^{i\theta}) \neq \mathbf{c}(d)</math> almost surely. This means that at least one of the events <math>\mathbf{c}(0) = \mathbf{c}(d)</math>, <math>\mathbf{c}(0) = \mathbf{c}(d e^{i\theta})</math> occurs with probability at most 1/2. The claim now follows from isometry invariance (3). <math>\Box</math>

=== Lemma 3 ===
(Using the K graph) We have
:<math>52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) + {\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} ) \geq 1 \quad (8).</math>

'''Proof''' Consider the 61-vertex graph <math>K</math> from [https://arxiv.org/abs/1804.02385 de Grey's paper]. It has 26 (isometric) copies of H, and thus 52 copies of the triangle <math>(1, e^{2\pi i/3}, e^{4\pi i/3})</math>. With probability at least <math>1 - 52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) </math>, none of these triangles are monochromatic. By the argument in that paper, this implies that the three linking diagonals <math>(-2, +2)</math>, <math>(-2 e^{2\pi i/3}, 2e^{2\pi i/3})</math>, <math>(-2 e^{4\pi i/3}, e^{-4\pi i/3})</math> are monochromatic. This gives the claim. <math>\Box</math>

=== Corollary 4 ===
(Existence of monochromatic <math>\sqrt{3}</math>-triangles) We have <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) \geq \frac{1}{104}</math>.

'''Proof''' The probability <math>{\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} )</math> is at most <math>{\bf P}( \mathbf{c}(-2) = \mathbf{c}(2)) = p_4</math>, which by Lemma 2 is at most 1/2. The claim now follows from Lemma 3. <math>\Box</math>

=== Computer-verified Claim 5 ===
(Using the graph M) One has <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = 0</math> (Note this contradicts Corollary 4).

'''Proof''' This simply reflects the fact that there is no 4-coloring of the 1345-vertex graph M from [https://arxiv.org/abs/1804.02385 de Grey's paper] with its central copy of H containing a monochromatic triangle. One can use other graphs for this purpose, such as the 278-vertex graph <math>M_1</math> or <math>V \oplus V \oplus H</math>. <math>\Box</math>

=== Computer-verified Claim 6 ===
(Using the graph <math>V \oplus V \oplus H</math>) One has <math>p_{8/3} = 1</math> (note this contradicts Lemma 2).

'''Proof''' This reflects the fact that every 4-coloring of <math>V \oplus V \oplus H</math> must assign the same color to 0 and 8/3. There is also a 745-vertex subgraph of <math>V \oplus V \oplus H</math> with the same property. <math>\Box</math>

=== Lemma 7 ===
(Using <math>G_{40}</math>) We have
:<math>59 p_{\sqrt{11/3}} + p_{8/3} \geq 1</math>.

'''Proof''' This reflects the fact that every 4-coloring of the 40-vertex graph <math>G_{40}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which none of the 59 pairs of vertices at distance <math>\sqrt{11/3}</math> apart, will assign the same color to 0 and 8/3. (This is presumably human-verifiable.) <math>\Box</math>

=== Corollary 8 ===
One has
:<math> p_{\sqrt{11/3}} \geq \frac{1}{118}</math>.

'''Proof''' Combine Lemma 7 and Lemma 2. <math>\Box</math>

=== Lemma 9 ===
(Using <math>G_{49}</math>) One has
:<math>18 {\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq p_{\sqrt{11/3}} .</math>

'''Proof''' This reflects the fact that every 4-coloring of the 49-vertex graph <math>G_{49}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which 0 and <math>\sqrt{11/3}</math> have the same color, at least one of the 18 copies of <math>(1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3)</math> is monochromatic. This is potentially human-verifiable. <math>\Box</math>

=== Corollary 10 ===
(Existence of monochromatic <math>1/\sqrt{3}</math>-triangles) One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq \frac{1}{2124}.</math>

'''Proof''' Combine Corollary 8 and Lemma 9. <math>\Box</math>

=== Computer-verified claim 11 ===
One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) = 0</math>. (This contradicts Corollary 10).

'''Proof''' This reflects the fact that the 627-vertex graph <math>G_{627}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] does not have any 4-colorings with <math>1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3</math> monochromatic. <math>\Box</math>

=== Lemma 12 ===
For certain special distances d, one can improve the bound in Lemma 2:

If <math>k \geq 1</math> is a natural number, <math>j\in\mathbb{Z}</math>, <math>\gcd(j,2k+1)=1</math>, and <math>r = \frac{1}{2} \csc\left(\frac{j\pi}{2k+1}\right)</math> then
:<math> p_r \leq \frac{k}{2k+1},</math>
thus for instance
:<math> p_{\frac{1}{\sqrt{3}}} \leq \frac{1}{3}.</math>

'''Proof''' Observe that the regular 2k+1-polygon <math>r, re^{2\pi i/(2k+1)}, r e^{4\pi i/(2k+1)}, \dots, r^{4k\pi i/(k+1)}</math> has unit side lengths. By the pigeonhole principle, we conclude that at most k of these vertices can have the same color as the origin, and the claim follows. <math>\Box</math>

On the other hand, for <math>k=2,j=1</math> we also know from the regular pentagon of unit sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}+1}{2}} \leq \frac{2}{5} \quad (9)</math>
since any 4-coloring of that pentagon has either one or two monochromatic diagonals.

Similarly, for <math>k=2,j=2</math> we also know from the regular pentagon of <math>\frac{\sqrt{5}-1}{2}</math> sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}-1}{2}} \leq \frac{2}{5}</math>
since any 4-coloring of that pentagon has either one or two monochromatic edges. More generally, if <math>a,b,c,d,e</math> are the diagonal lengths of a pentagon with unit sides, then
:<math> 1 \leq p_a + p_b + p_c + p_d + p_e \leq 2.</math>

=== Lemma 13 ===
We have
:<math> 7 p_{\frac{1}{\sqrt{3}}} \geq p_{\sqrt{3}}</math>.

'''Proof''' Consider the unit rhombus <math>0, 1, e^{i\pi/3}, e^{-i\pi/3}</math> together with the centers <math>e^{i\pi/6}/\sqrt{3}, e^{-i\pi/6}/\sqrt{3}</math>. With probability <math>p_{\sqrt{3}}</math>, the two far vertices <math>e^{i\pi/3}, e^{-i\pi/3}</math> are the same color, and then 0,1 will be two other colors. This forces either one of the centers <math>e^{i\pi/6}/\sqrt{3}</math> of a triangle to have a common color with one of the vertices of that triangle, or the two centers must have the same color. Thus in any event one of the seven edges of distance <math>1/\sqrt{3}</math> is monochromatic, giving the claim. <math>\Box</math>

=== Corollary 14 ===
We have <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{728}</math>.

This slightly improves upon the lower bound of 1/2124 coming from Corollary 10.

'''Proof''' Combine Corollary 4 and Lemma 13. <math>\Box</math>

=== Lemma 15 ===
One has
:<math>p_{\sqrt{3}} + p_2 \geq \frac{2}{3}</math> and <math>2 p_{\sqrt{3}} + p_2 \geq 1</math>.

'''Proof''' As noted in de Grey's paper, there are essentially four 4-colorings of H. H has six edges of length <math>\sqrt{3}</math> and three of length <math>2</math>. If we let a denote the number of monochromatic <math>\sqrt{3}</math> edges and b the number of monochromatic <math>2</math>-edges, we see from inspection of all four colorings that <math>(a,b)</math> is either <math>(6, 0), (4,0), (2, 1)</math>, or <math>(0,3)</math>. In particular, one always has <math>\frac{a}{6} + \frac{b}{3} \geq \frac{2}{3}</math> and <math>2\frac{a}{6} + \frac{b}{3} \geq 1</math>. Taking expectations, we obtain the claim. <math>\Box</math>

=== Corollary 16 ===
We have <math>p_2 \geq \frac{1}{6}</math>, <math>p_{\sqrt{3}} \geq \frac{1}{4} </math> and <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{28}</math>.

'''Proof''' Combine Lemma 2, Lemma 15, and Lemma 13. <math>\Box</math>

=== Lemma 17 ===
If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths a,b,c. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also, thus
:<math>{\bf P}(\mathbf{c}(0) \neq \mathbf{c}(a)) + {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(b)) \geq {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(c))</math>
and the claim follows. <math>\Box</math>

Note that Lemma 2 follows from the a=b, c=1 case of this lemma. Iterating this lemma starting with Lemma 2 we can also obtain slightly nontrivial upper bounds on <math>p_a</math> for small values of a, e.g. <math>p_a \leq 1 - 2^{-k}</math> when <math>a \geq 2^{-k}, k\in\mathbb{Z}^+</math>.

Further, we can generalise the a=b case to one in which the triangle is replaced by a (k+1)-gon of which one edge is 1 and the others are all equal, leading to the stronger result <math>p_a \leq 1 - 1/k</math> when <math>a \geq 1/k, k\in\mathbb{Z}^+ \land k>1</math>. Further strengthening is achieved by using <math>1/\sqrt{3}</math> as the long edge, given Lemma 12.

=== Lemma 18 ===
Whenever <math>d>0</math>, one has the inequalities
:<math> |p_{\phi d} - p_d| \leq \frac{2}{5}, p_{\phi d} + p_d \geq \frac{1}{5}, 2p_d - p_{\phi d} \leq 1, 2 p_{\phi d} - p_d \leq 1</math>
where <math>\phi := \frac{1+\sqrt{5}}{2}</math> is the golden ratio. Also we have
:<math> p_{d/\sqrt{3}} \leq \frac{1}{3} + p_d, \frac{1}{2} + \frac{1}{2} p_d.</math>

Note that this generalises (9), as well as a special case of Lemma 12.

'''Proof''' Consider the regular pentagon with sidelength <math>d</math>, so it also has 5 diagonals of length <math>\phi d</math>. Let <math>a \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic edges and let <math>b \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic diagonals. Observe:
* <math>a,b</math> cannot both be zero (pigeonhole principle).
* <math>a</math> cannot be 4. Similarly, <math>b</math> cannot be 4.
* If <math>a=5</math>, then <math>b=5</math>, and conversely.
* If <math>a=0</math>, then <math>b=1,2</math>; similarly, if <math>b=0</math>, then <math>a=1,2</math>.

From this we observe the inequalities
:<math> |\frac{a}{5}-\frac{b}{5}| \leq \frac{2}{5}; \frac{a}{5} + \frac{b}{5} \geq \frac{1}{5}; 2 \frac{a}{5} - \frac{b}{5} \leq 1; 2\frac{b}{5} - \frac{a}{5} \leq 1</math>
and on taking expectations we obtain the first claim. Similarly, if one considers the colorings of an equilateral triangle of sidelength <math>d</math> together with its center, and counts the numbers <math>a,b \in \{0,1,2,3\}</math> of monochromatic edges of length <math>d</math> and <math>d/\sqrt{3}</math> respectively, one observes that one always has <math>\frac{b}{3} \leq \frac{1}{3} + \frac{2}{3} \frac{a}{3}, \frac{1}{2} + \frac{1}{2} \frac{a}{3}</math>, and on taking expectations one obtains the claim.<math>\Box</math>

The hexagon <math>H</math> has essentially four distinct colorings: the coloring <math>\hbox{2tri}</math> with two triangles, the coloring <math>\hbox{1tri}</math> with one triangle, the coloring <math>\hbox{axisym}</math> that is symmetric around an axis, and the coloring <math>\hbox{centralsym}</math> that is symmetric around the central point. This gives four probabilities <math>p_{H = 2tri}, p_{H = 1tri}, p_{H = axisym}, p_{H = centralsym}</math> that sum to 1. By counting the number of monochromatic edges of length <math>\sqrt{3}, 2</math> respectively, one also obtains the identities
:<math>p_{\sqrt{3}} = p_{H = 2tri} + \frac{2}{3} p_{H = 1tri} + \frac{1}{3} p_{H = axisym}; \quad p_2 = \frac{1}{3} p_{H=axisym} + p_{H=centralsym}</math>
which reproves Lemma 15. Also
:<math> {\bf P}( \mathbf{c}(0) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = p_{H = 2tri} + \frac{1}{2} p_{H=1tri}.</math>
Any 4-coloring of L contains at least one triangle within one of its 52 copies of H, thus
:<math> p_{H = 2tri} + \frac{1}{2} p_{H=1tri} \geq \frac{1}{52}</math>
which reproves Corollary 4.

=== Lemma 19 ===
(Hubai) One has <math>p_{H = 1tri} + p_{H = axisym} \geq \frac{1}{10}</math>.

'''Proof''' Consider five copies of H centred at 0,1,2,3,4. With probability at least <math>1 - 5( p_{H = 1tri} + p_{H = axisym} )</math>, none of these copies of H are colored 1tri or axisym, and so must be colored 2tri or centralsym. One can check then that if one of the copies is colored 2tri, then so is any adjacent copy; thus all five copies are colored 2tri, or all five are colored centralsym. In either case we see that -1 and 5 are colored the same color. Comparing with Lemma 2 then gives the claim. <math>\Box</math>

=== Theorem 20 ===
One has <math>p_{H = 1tri} > 0</math>.

'''Proof''' Suppose for contradiction that <math>p_{H = 1tri} = 0</math>. One can then run a version of the de Bruijn-Erdos argument to obtain a coloring in which 1tri hexagons are completely nonexistent (since there are arbitrarily large finite colorings with this property). Consider the triangular lattice <math>{\bf Z}[e^{\pi i/3}]</math>. We 2-color the edges of this lattice by coloring an edge black if it is the short diagonal of a unit rhombus with monochromatic long diagonal, and white otherwise. The four colorings of hexagons lead to four possible colorings at each vertex:

* If H is colored 2tri, then all six edges to the centre of H are black.
* If H is colored 1tri, then two edges to the centre of H at 120 degree angles are white, the other four are black.
* If H is colored axisym, then two opposing edges of the centre of H are black, the other four are white.
* If H is colored centralsym, then all six edges to the centre of H are black.

In particular, as we are assuming no 1tri hexagons, the faces cut out by the black edges have angles 60 degrees, and thus must be equilateral triangles, sectors of angle 60, half-planes, or the entire plane. If there is at least one equilateral triangle, then the rest of the black edges must form an equilateral lattice with that triangle sidelength. This leads to only a small number of possible hexagon colorings in the lattice:

# Case 1: All edges white.
# Case 2: All edges black.
# Case 3.k: For some natural number <math>k \geq 2</math>, the length k edges joining adjacent vertices in some coset of <math>k \cdot {\mathbf Z}[ e^{\pi i/3} ]</math> are all black, and the remaining edges are white.
# Case 4: Each horizontal row consists either entirely of black edges, or entirely of white edges.
# Case 5: Each northwest row consists either entirely of black edges, or entirely of white edges.
# Case 6: Each northeast row consists either entirely of black edges, or entirely of white edges.
# Case 7: Six rays of black edges meeting at a common vertex; all other edges white.

Technically, Case 1 is contained in Cases 4,5,6 as written above, but this will not be an issue. One can view Case 7 as a limiting case <math>k=\infty</math> of Case 3.k; Case 2 is similarly the opposite limiting case <math>k=1</math>.

In the first case, the coloring is periodic with periods <math>2, 2 e^{\pi i/3}</math>. In the second case, it is periodic with periods <math>3, 3 e^{\pi i/3}</math>. In the third case, it is periodic with periods <math>3k, 3k e^{\pi i/3}</math>. Also note that for each k, one can check if Case 3.k holds by inspecting the coloring at a finite number of vertices. Thus the event that Case 3.k holds is "measurable" in the sense that a meaningful probability can be assigned. (But Cases 1,2,4,5,6 are not measurable events, they require an infinite number of points to be inspected, and the probability measure we are using is only finitely additive rather than infinitely additive.) In Case 4, the coloring is periodic with period 2; also, every coset of <math>2 \cdot {\bf Z}[e^{\pi i/3}]</math> is 2-colored. Similarly for Case 5 and 6 (where the periods are <math>2 e^{2\pi i/3}</math> and <math>2 e^{4\pi i/3}</math> respectively.)

Let <math>\alpha_k</math> be the probability that Case 3.k holds for the given value of <math>k</math>. Then <math> \sum_{k=2}^K \alpha_k \leq 1</math> for any k, hence <math>\sum_{k=2}^\infty \alpha_k \leq 1</math>. In particular, we can find <math>K_1</math> such that <math>\sum_{k={K_1}}^\infty \alpha_k \leq 0.1</math> (say). Let <math>P</math> be six times the least common multiple of <math>1,2,\dots,K_1</math>. Then the coloring is P- and <math>P e^{\pi i/3}</math>-periodic for Case 1, Case 2, and all Case 3.k with <math>k \leq K_1</math>. On the other hand, if <math>K_2</math> is sufficiently large depending on <math>P</math>, and Case 3.k holds for some <math>k \geq K_2</math>, then almost all of the hexagons are colored centralsym, which makes the coloring "almost <math>P, P e^{\pi i/3}</math>-periodic" in the sense that
:<math>{\bf c}(z+P e^{\pi i j/3}) = {\bf c}(z) \hbox{ for } j=0,1,2,3,4,5</math>
will hold for at least <math>0.9</math> of the lattice points <math>z \in {\bf Z}[e^{\pi i/3}]</math> with <math>|z| \leq K_2</math>. Similarly for Case 7 (which is sort of a <math>k=\infty</math> limiting case of Case 3.k.) Thus, with the probability <math> \geq 1 - \sum_{k=K_1}^{K_2} \alpha_k \geq 0.9</math>, the coloring of the seven vertices <math>{\bf c}(0), {\bf c}(P e^{\pi ij/3}, j=1,\dots,6</math> is (up to rotation and recoloring) one of the three patterns of the central and linking vertices in Figure 3 of Aubrey's paper, namely

* <math>{\bf c}(0) = {\bf c}(P) = {\bf c}(P e^{\pi i/3}) = {\bf c}(P e^{2\pi i/3}) = {\bf c}(P e^{3\pi i/3}) = {\bf c}(P e^{4\pi i/3}) = {\bf c}(P e^{5\pi i/3}) </math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{3\pi i/3})</math>; <math>{\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{3\pi i/3})</math>; <math> {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>.

Using the spindling argument from Aubrey's paper, we conclude that the third possibility must in fact hold with probability at least 0.8; on the other hand, from Lemma 2 this scenario can only occur with probability at most 1/2, giving the required contradiction.
<math>\Box</math>

One should be able to refine this argument to show that <math>p_{H = 1tri} > c</math> for an absolute constant <math> c>0 </math>.

=== Lemma 21 ===
Providing a tighter bound for Lemma 17 with a more thorough proof: If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths <math>a,b,c</math>. Let <math>\left|z_2\right|=b,\left|a-z_2\right|=c</math>. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also: <math>\mathbf{c}(a)\neq\mathbf{c}(z_2)\Rightarrow[\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)]</math>. <math>[A\Rightarrow B]\Rightarrow {\bf P}(A)\leq{\bf P}(B)</math> thus
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) \geq {\bf P}(\mathbf{c}(a) \neq \mathbf{c}(z_2)) = 1-p_c</math>
<math>{\bf P}(A\lor B) +{\bf P}(A\land B)={\bf P}(A)+{\bf P}(B)</math>, so
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)) + {\bf P}(\mathbf{c}(0)\neq\mathbf{c}(z_2)) - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>1-p_c \leq 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
Using the law of cosines: <math>z_2=b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)</math>
:<math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math> <math>\Box</math>

=== Lemma 22 ===

One has <math>3 p_{1/\sqrt{3}} \geq {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) )</math>.

'''Proof''' Let <math>z</math> be a complex number of magnitude <math>1/\sqrt{3}</math> that is a unit distance from 1. If <math>\mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) = c</math> (say), then <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> cannot be colored with <math>c</math>; also, <math>z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> are the vertices of a unit equilateral triangle and thus must take on three different colors. By the pigeonhole principle, one of <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> must then take the same color as the origin, and the claim follows. <math>\Box</math>

=== Lemma 23 ===

One has <math>4 p_{(\sqrt{6} \pm \sqrt{2})/2} + p_{\sqrt{2}} \geq 1</math>. In particular, by Lemma 2, <math>p_{(\sqrt{6}+\sqrt{2})/2} \geq 1/8</math>.

'''Proof''' [ExIs2018b] We just prove the claim for the + sign (the - sign can then be obtained after applying the Galois conjugacy that maps <math>\sqrt{3}</math> to <math>-\sqrt{3}</math>, leaving <math>\sqrt{2}</math> unchanged). Set <math>d := \frac{\sqrt{6}+\sqrt{2}}{2}</math>, and consider the five vertices

:<math>0, e^{5\pi i/4}, e^{5\pi i/4} + d, e^{5\pi i/4} + e^{\pi i/3} d, e^{5\pi i/4} + (e^{\pi i/3}-i)d.</math>

One can check that of the ten edges determined by these five vertices, five have unit length, four have length d, and the remaining distance (from 0 to <math>e^{5\pi i/4}+d</math>) has distance <math>\sqrt{2}</math>. Since <math>\mathbf{c}</math> must make one of the latter five edges monochromatic, the claim follows. <math>\Box</math>

=== Lemma 24 ===

One has <math>p_{\sqrt{2}} \geq \frac{1}{14}</math>.

'''Proof''' In page 7 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 20 unit distance edges and 14 edges of length <math>\sqrt{2}</math> is constructed. The coloring <math>\mathbf{c}</math> must make one of the 14 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 25 ===

Let <math>d = \frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math> and <math>e = \frac{3^{1/4} \sqrt{2} + \sqrt{3} - 1}{2}</math>.
Then one has <math>14 p_d + p_e \geq 1</math>. In particular, by Lemma 2, <math>p_d \geq 1/28</math>.

'''Proof''' In page 9 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 19 unit edges, 14 edges of length d, and one edge of length e is constructed. The coloring <math>\mathbf{c}</math> must make one of the 15 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 26 ===

Let <math>d = \sqrt{3/2 + \sqrt{33}/6}</math>. Then <math>7 p_d \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_d \geq \frac{1}{196}</math>.

'''Proof''' In page 11 of [ExIs2018b], a graph of nine vertices consisting of 12 unit edges and 7 edges of length d is constructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Thus, <math>\mathbf{c}</math> can only make the AB edge monochromatic if one of the seven length d edges is monochromatic. The claim follows.
<math>\Box</math>

=== Lemma 27 ===

One has <math>27 p_{\sqrt{5/3}} \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_{\sqrt{5/3}} \geq \frac{1}{756}</math>.

'''Proof''' In page 13 of [ExIs2018], a graph of 33 vertices with some unit edges and 27 edges of length <math>\sqrt{5/3}</math> is contructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Now repeat the proof of Lemma 26. <math>\Box</math>

=== Computer-verified claim 28 ===

One has <math>p_{2/\sqrt{3}} \geq \frac{1}{177}</math>.

'''Proof''' In page 15 of [ExIs2018], a 5-chromatic graph of 103 vertices, 312 unit edges, and 177 edges of length <math>2/\sqrt{3}</math> is constructed. <math>\mathbf{c}</math> must make one of the latter edges monochromatic, giving the claim. <math>\Box</math>

=== Lemma 29 ===

One has <math>p_{(\sqrt{6} \pm \sqrt{2})/2} \geq 1/6</math> (this improves the bound in Lemma 23).

'''Proof''' Use graphs 505 and 507 from [S2004] and the spindle bound. <math>\Box</math>

=== Lemma 30 ===
For <math>m > n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' <math>n</math> colors and <math>m</math> points necessitates at least 2 having equal color. I.e.

:<math>{\bf P}\left(\bigvee_{k=0}^n \bigvee_{j=k+1}^n\ \mathbf{c}(z_k) = \mathbf{c}(z_j)\right) = 1</math>

The lemma then follows immediately from the fact:
:<math>{\bf P}\left(\bigcup_{k} E_k\right) \leq \sum_{k} {\bf P}\left(E_k\right) \,\Box</math>

=== Corollary 31 ===
Let <math>\lvert z_k\rvert=1</math>. Then for <math>m \geq n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' Use lemma 30 on the set <math>\left\{z_k \bigg\vert 1\leq k\leq m \land k\in\mathbb{Z}\right\}\cup\{0\}</math>. Simplify using <math>{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(0) \right)=0</math>.
<math>\Box</math>

=== Lemma 32 ===
For <math>x\in\mathbb{R}</math> and an <math>n</math>-coloring of the plane, <math>\sum_{k=1}^{n-1}\left(n-k\right){\bf P}\left(\mathbf{c}\left(0\right) = \mathbf{c}\left( 2\sin\left(\frac{kx}{2}\right) \right) \right) \geq 1</math>.

'''Proof''' Use corollary 31 on the set <math>\left\{e^{ikx} \bigg\vert 0\leq k < n \land k\in\mathbb{Z}\right\}</math>. and simplify by grouping lengths.
<math>\Box</math>

=== Corollary 33 ===
Interesting(easy to simplify results of) values for <math>x</math> in Lemma 32 are in <math>\left\{x \bigg\vert \sin\left(\frac{kx}{2}\right)=1 \land 1\leq k < n \land k\in\mathbb{Z}\right\}</math>.
For 4-colorings, this gives
:<math>2p_{\sqrt 3}+p_2 \geq 1</math>
:<math>3p_{(\sqrt 3-1)/\sqrt 2}+p_{\sqrt 2} \geq 1</math>
:<math>3p_{2\sin(\pi/18)}+2p_{2\sin(\pi/9)} \geq 1</math>

=== Lemma 34 ===
Generalizing the note of Lemma 17, <math>\lvert d_1\rvert= d_1 > \lvert d_0\rvert= d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>

'''Proof''' let <math>\lvert z_{j+1} -z_j\rvert=d_0 > 0, \lvert z_{j+n} -z_0\rvert=d_1>0</math>.

Base case, <math>n=2</math>, by Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle:
:<math>2d_0\geq d_1\Rightarrow 2p_{d_0}\leq 1+p_{d_1}</math>
The inductive step is Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle. After induction:
:<math>[n\geq 2\land nd_0\geq d_1]\Rightarrow np_{d_0}\leq n-1+p_{d_1}</math>
Substitute <math>n=\left\lceil\frac{d_1}{d_0}\right\rceil</math>, simplify, rearrange:
:<math>d_1 > d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>
<math>\Box</math>

=== Proposition 35 ===

If <math>d > 1/\sqrt{2}</math> obeys the inequalities
:<math>\sin( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
and
:<math>\cos( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
then
:<math>p_d \leq \frac{1}{2} - \frac{1}{188}.</math>

(One can check that the conditions are obeyed precisely when <math>d \geq \frac{\sqrt{33}-1}{8} = 0.84307\dots</math>.)

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. Since <math>AB</math> is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the triangle <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>.

Now let <math>BADE</math> be a rhombus with sidelengths d and <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. By the hypotheses, the diagonals BD, AE of this rhombus have length at least 1/2, and hence are monochromatic with probability at most 1/2 by Lemma 2. As above, ABD and BDE are each monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>. As BD is monochromatic with probability at most <math>1/2</math>, we conclude that BADE is monochormatic with probability at least <math>\frac{1}{2}-6\delta</math>.

Now let <math>EDFG</math> be another rhombus congruent to <math>BADE</math>. As BD, AE have length at least 1/2, at least one of the long diagonals BF, AG have length at least 1/2 (the diagonal opposite an obtuse or right-angled triangle will work). Let's say BF has length at least 1/2. As BADE and EDFG are both monochromatic with probability at least <math>\frac{1}{2}-6\delta</math>, and the common edge DE is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the entire configuration ABDEFG is monochromatic with probability at least <math>\frac{1}{2}-11\delta</math>. In particular the pentagon ABDEF is monochromatic with at least this probability. However, in this pentagon, the five edges BA, AD, DE, EB, EF are monochromatic with probability at most <math>\frac{1}{2}-\delta</math>, and the other five edges are monochromatic with probability at most <math>\frac{1}{2}</math> by Lemma 2. Thus the probability that at least one of the edges of this pentagon is monochromatic is at most <math>(\frac{1}{2}-11\delta) + 5 \times 10\delta + 5 \times 11\delta = \frac{1}{2}+94\delta</math>. On the other hand, by the pigeonhole principle, this probability is 1. The claim follows. <math>\Box</math>

=== Proposition 36 ===

If <math>d \ge \frac{2}{\sqrt{15}} = 0.5163\dots</math>, then <math>p_d \leq \frac{1}{2} - \frac{1}{62}</math>,

if <math>d \ge \frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math>, then <math>p_d \leq 0.48</math>,

and <math>\limsup_d p_d\leq \frac{323}{675}=0.4785\ldots</math> (so <math>p_d<0.4786</math> if <math>d</math> is large enough).

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. A simple calculation shows that if <math>d \ge \frac{2}{\sqrt{15}}</math>, then <math>|BD| \ge \frac{1}{2}</math>.
If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. By inclusion-exclusion, we conclude that outside of the event that <math>AB</math> is monochromatic, the probability that <math>AD</math> is monochromatic is at most <math>2\delta</math>.

Now let <math>ADE</math> the <math>180^\circ</math> rotation of <math>ADB</math> around the midpoint of <math>AD</math>, and let <math>FDE</math> be the <math>180^\circ</math> rotation of <math>ADE</math> around the midpoint of <math>DE</math>. By the hypotheses, the line segments <math>AE, BD, BE, BF, DF</math> all have length at least 1/2. Let <math>{\mathcal E}</math> be the event that at least one of <math>AB, AD, DE, EF</math> is monochromatic. By the previous paragraph, this event occurs with probability at most <math>\frac{1}{2}-\delta+2\delta+2\delta+2\delta = \frac{1}{2}+5\delta</math>.

By previous considerations, <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>, and this event lies in <math>{\mathcal E}</math>. On the other hand, <math>BD</math> is monochromatic with probability at most 1/2 by Lemma 2. We conclude that outside of <math>{\mathcal E}</math>, <math>BD</math> is only monochromatic with probability at most <math>3\delta</math>. A similar argument (replacing <math>ABD</math> by <math>DAE</math> or <math>EDF</math>) shows that outside of <math>{\mathcal E}</math>, <math>AE</math> is monochromatic with probability at most <math>3\delta</math>, and similarly for <math>DF</math>. Now we consider <math>EB</math>. By previous considerations, the probability that <math>ABDE</math> is monochromatic is at least <math>\frac{1}{2}-5\delta</math>, and this event lies inside <math>{\mathcal E}</math>. Thus, outside of <math>{\mathcal E}</math>, the probability that <math>EB</math> is monochromatic is at most <math>5\delta</math>; similarly for <math>AF</math>. Finally, the probability that <math>BF</math> is monochromatic outside of <math>{\mathcal E}</math> is at most <math>7\delta</math>. We conclude that outside of an event of probability
:<math>\frac{1}{2}+5\delta + 3\delta+3\delta+3\delta+5\delta+5\delta+7\delta = \frac{1}{2} + 31\delta,</math>
none of the ten edges connecting <math>A,B,D,E,F</math> are monochromatic. But by the pigeonhole principle, this cannot occur in a 4-coloring, hence <math>\frac{1}{2} + 31 \delta \geq 1</math>, and the first claim follows.

For the second claim, we need to use an iterative argument, by feeding the bounds obtained back into the place in the proof where Lemma 2 is currently invoked. To have all occurring distances stay larger than <math>d</math>, we only need to check <math>|BD| \ge d</math>. Equality occurs when <math>ABD</math> is an equilateral triangle, which means that <math>ABC</math> and <math>ACD</math> are isosceles triangles with sides <math>d,d,1</math> and either with angles <math>150^\circ,15^\circ,15^\circ</math>, or with angles <math>30^\circ,75^\circ,75^\circ</math>. From here calculation gives <math>d \ge \frac{1}{2sin(75^\circ)}=\frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math> and <math>d \le \frac{1}{2sin(15^\circ)}=\frac{\sqrt{3}+1}{\sqrt{2}}=1.9318\dots</math>, but the upper bound is not really important, as for us it is enough that <math>|BD|</math> always stay above <math>d_0=\frac{\sqrt{3}-1}{\sqrt{2}}</math>, which occurs everywhere above this value. Now pick a <math>d\ge d_0</math> for which <math>p_d\ge \frac{1}{2}-\delta-\varepsilon</math>, where <math>\sup_{d\ge d_0} p_d= \frac{1}{2}-\delta</math> and <math>\varepsilon</math> is a small positive number. The calculation of the first case gives <math>\frac{1}{2}+5\delta + 2\delta+2\delta+2\delta+4\delta+4\delta+6\delta+O(\varepsilon) =\frac{1}{2} + 25 \delta+O(\varepsilon) \geq 1</math>, from which <math>\delta\ge 0.02</math> if we choose <math>\varepsilon</math> small enough.

To prove the last claim, we modify the construction; we obtain <math>F</math> by reflecting <math>B</math> to <math>AD</math>, to win <math>2\delta</math> in the last step of the calculation. To invoke Lemma 2, we need (among other things) that <math>EF</math> is at least 1/2, and to iterate in a straight-forward way, we would need a value <math>d_0</math> for which <math>EF</math> is at least <math>d_0</math> if <math>AB</math> is <math>d\ge d_0</math>, but such a <math>d_0</math> doesn't exist. We can, however, still iterate in a weaker sense, as <math>6</math> of the occurring <math>10</math> distances tend to infinity as <math>d=|AB|</math> tends to infinity, and the remaining <math>4</math> are also larger than <math>\frac{\sqrt{3}-1}{\sqrt{2}}</math>, so their probability of them being monochromatic is at most <math>0.48=(0.5-\delta)+(\delta-0.02)</math>. What we get eventually is <math>\frac{1}{2} + 25 \delta-2\delta+ 4(\delta-0.02)+O(\varepsilon) =0.42 + 27 \delta+O(\varepsilon) \geq 1</math>, from which <math>p_d\le \frac{323}{675}=0.4785\ldots</math> for <math>d</math> large enough.
<math>\Box</math>

=== Lemma 37 ===

One has <math>\sup_{0 < d < 2} p_d \geq 1/3</math>.

'''Proof''' For a large integer <math>n</math>, consider the points <math>e^{2\pi i j/n}</math> with <math>j=1,\dots,n</math>. Any unit distance coloring will color these points in at most 3 colors, hence divides the n points into three color classes of some size <math>n_1,n_2,n_3</math>. The number of monochromatic pairs is then
:<math>\frac{n_1(n_1-1)}{2} + \frac{n_2(n_2-1)}{2} + \frac{n_3(n_3-1)}{2} = \frac{1}{2} (n_1^2+n_2^2+n_3^2) + O(n) \geq \frac{1}{6} n^2 + O(n)</math>
by Cauchy-Schwarz. Thus at least <math>1/3-O(1/n)</math> of the pairs are monochromatic. Taking expectations and using the pigeonhole principle, we conclude that one of the distances has a probability at least <math>1/3 -O(1/n)</math> of being monochromatic, giving the claim. <math>\Box</math>

=== Lemma 38 ===

Let ABC be a unit-edge equilateral triangle, and let D be an arbitrary point. Let <math>|AD|, |BD|</math> and <math>|CD|</math> be <math>x,y,z</math> respectively. Then <math>p(x)+p(y)+p(z) \leq 1</math>. In particular, examining the case e=f, if <math>p(d) \geq k</math> then <math>p(\sqrt(d(d \pm \sqrt 3) + 1) \leq (1-k)/2</math>.

'''Proof''' At most one of <math>AD, BD</math> and <math>CD</math> can be monochromatic. <math>\Box</math>

Note: A consequence is that a 4-chromatic unit-distance graph G can demonstrate CNP <math>> 4</math> if, for the {x,y,z} arising from some choice of D above, G contains three equal-sized non-empty sets v_x, v_y, v_z of vertex-pairs such that (a) each vertex-pair within v_x is at distance x (resp. y and z), and (b) in any 4-colouring of G, more than 1/3 of the vertex-pairs in the union of the three sets are monochromatic. Note that this demonstration does not require that v_x contain all the vertex-pairs of G that are at distance x (resp. y and z), nor even that the graph {A,B,C,D} which gives rise to {x,y,z} be a subgraph of G. It seems plausible to find such a graph that is small (and/or symmetrical) enough that its colourings can be human-analysed to establish this property.

== Simplification rules for triplets of points in the complex plane ==
Deduced from the rule <math>{\bf P}(A\land B)+{\bf P}(A\land \lnot B)={\bf P}(A)</math>.
:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) = {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) - {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) ) - {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) \neq {\mathbf c}(z_0) ) + {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) = {\mathbf c}(z_0) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

== Proofs of bounds for conditional probabilities ==
The trivial case, valid where <math>\left|d\right|\neq 1</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) = {\mathbf c}(d) )=0</math>

Trivial plus Baye's Theorem, valid where <math>d\neq 0</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) )=\frac{{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )}\leq 1</math>
Rearrange:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )+{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1</math>

Spindle method: for <math>\left|d\right|=d\geq 1/2</math> and <math>\theta=2\text{arcsin}\left(\frac{1}{2d}\right)</math>:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{i\theta}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) ) = \frac{1}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )} - 1\leq 1</math>
which is another way to see <math>\left|d\right|=d\geq 1/2\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1/2</math>.

== Further questions ==
* For <math>n,m\geq CNP</math>, what consistent relationships exist between <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert n\text{ colors}\right)</math> and <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert m\text{ colors}\right)</math>? How can these relationships be used to sharpen arguments of the probabilistic formulation?

[[Category:Polymath16]]

Probabilistic formulation of Hadwiger-Nelson problem

2018-07-15T23:41:43Z

Aubrey: /* Lemma 38 */

Suppose for sake of contradiction that we have a 4-coloring <math>c: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with no unit edges monochromatic, thus

:<math>c(z) \neq c(w) \hbox{ whenever } |z-w| = 1. \quad (1)</math>

We can create further such colorings by composing <math>c</math> on the left with a permutation <math>\sigma \in S_4</math> on the left, and with the (inverse of) a Euclidean isometry <math>T \in E(2)</math> on the right, thus creating a new coloring <math>\sigma \circ c \circ T^{-1}: {\bf C} \to \{1,2,3,4\}</math> of the complex plane with the same property. This is an action of the solvable group <math>S_4 \times E(2)</math>.

It is a fact that all solvable groups (viewed as discrete groups) are [https://en.wikipedia.org/wiki/Amenable_group amenable], so in particular <math>S_4 \times E(2)</math> is amenable. This means that there is a finitely additive probability measure <math>\mu</math> on <math>S_4 \times E(2)</math> (with all subsets of this group measurable), which is left-invariant: <math>\mu(gE) = \mu(E)</math> for all <math>g \in S_4 \times E(2)</math> and <math>E \subset S_4 \times E(2)</math>. This gives <math>S_4 \times E(2)</math> the structure of a finitely additive probability space. We can then define a random coloring <math>{\mathbf c}: {\bf C} \to \{1,2,3,4\}</math> by defining <math>{\mathbf c} := {\mathbf \sigma} \circ c \circ {\mathbf T}^{-1}</math>, where <math>({\mathbf \sigma},{\mathbf T})</math> is the element of the sample space <math>S_4 \times E(2)</math>. Thus for any complex number <math>z</math>, the random color <math>{\mathbf c}(z)</math> is a random variable taking values in <math>\{1,2,3,4\}</math>. The left-invariance of the measure implies that for any <math>(\sigma,T) \in S_4 \times E(2)</math>, the coloring <math> \sigma \circ {\mathbf c} \circ T^{-1}</math> has the same law as <math>{\mathbf c}</math>. This gives the color permutation invariance

:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(z_1) = \sigma(c_1), \dots, {\mathbf c}(z_k) = \sigma(c_k) )\quad (2)</math>

for any <math>z_1,\dots,z_k \in {\bf C}</math>, <math>c_1,\dots,c_k \in \{1,2,3,4\}</math>, and <math>\sigma \in S_4</math>, and the Euclidean isometry invariance
:<math> {\bf P}( {\mathbf c}(z_1) = c_1, \dots, {\mathbf c}(z_k) = c_k ) = {\bf P}( {\mathbf c}(T(z_1)) = c_1, \dots, {\mathbf c}(T(z_k)) = c_k. \quad (3)</math>
(In probabilistic language, this means that the random coloring is a [https://en.wikipedia.org/wiki/Stationary_process stationary process] with respect to the action of <math>S_4 \times E(2)</math>. The extraction of a stationary process from a deterministic object is an example of the ''Furstenberg correspondence principle''.)

== Bounds on <math>p_d</math> for 4-colourings ==

A class of correlations that is of particular interest is that of vertex pairs at some distance <math>d</math>. Accordingly, define
:<math>p_d := {\bf P}( \mathbf{c}(0) = \mathbf{c}(d) ).</math>

{| border=1
|-
! distance !! Lower bound !! Lower-bounding graph/method !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math> \geq 1/2</math>
|
|
| 1/2
| Spindle
|
|-
| <math>\geq \frac{2}{\sqrt{15}}</math>
|
|
| <math>\frac{1}{2} - \frac{1}{62}</math>
| Proposition 36
|
|-
| <math>\geq \frac{\sqrt{3}-1}{\sqrt{2}}</math>
|
|
| <math>0.48</math>
| Proposition 36
|
|-
| large enough
|
|
| <math>\frac{323}{675}=0.4785\ldots</math>
| Proposition 36
|
|-
| <math> d\geq 1/n, n>1</math>
|
|
| <math>1-\frac{1}{n}</math>
| (n+1)-gon with one edge length 1 and the rest d, Lemma 34
|
|-
| <math> d\geq 1/(n \sqrt{3}), n>1</math>
|
|
| <math>(3n-2)/3n</math>
| (n+1)-gon with one edge length <math>1/\sqrt{3}</math> and the rest d, Lemma 34
| Not better than the above on intervals <math>\left(\frac{1}{7},\frac{1}{4\sqrt{3}}\right),\left(\frac{1}{4},\frac{1}{2\sqrt{3}}\right)</math>
|-
| 1
| 0
| Unit edge
| 0
| Unit edge
|
|-
| <math>\frac{1}{\sqrt{3}}</math>
| 1/28
| Unit diamond plus centres of triangles, together with H, Corollary 16
| 1/3
| Unit triangle plus its centre
|
|-
| 2
| 1/4
| <math>G_{21}</math>
| <math>0.48</math>
| Proposition 36
| Lower bound computer verified
|-
| <math>{\sqrt{3}}</math>
| 1/4
| H, Corollary 16
| <math>0.48</math>
| Proposition 36
|
|-
| <math>{\frac{\sqrt{5}+1}{2}}</math>
| 1/5
| regular pentagon with unit side length
| 2/5
| regular pentagon with unit side length
|
|-
| <math>{\frac{\sqrt{5}-1}{2}}</math>
| 1/5
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
| 2/5
| regular pentagon with <math>{\frac{\sqrt{5}-1}{2}}</math> side length
|
|-
| <math>{\sqrt{11/3}}</math>
| 1/118
| <math>G_{40}</math>
| <math>0.48</math>
| Proposition 36
| computer-verified
|-
| 8/3
| 1
| <math>V \oplus V \oplus H</math>
| <math>0.48</math>
| Proposition 36
| computer-verified; leads to contradiction
|-
| <math>\frac{\sqrt{6} \pm \sqrt{2}}{2}</math>
| 1/6
| An arrangement of five vertices
| <math>0.5</math> and <math>0.48</math>
| Proposition 36
|
|-
| <math>\sqrt{2}</math>
| 1/14
| A graph of 13 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>\frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math>
| 1/28
| A graph of 13 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>\sqrt{3/2 + \sqrt{33}/6}</math>
| 1/196
| A graph of 9 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>\sqrt{5/3}</math>
| 1/756
| A graph of 33 vertices
| <math>0.48</math>
| Proposition 36
|
|-
| <math>2/\sqrt{3}</math>
| 1/177
| A graph of 103 vertices
| <math>0.48</math>
| Proposition 36
| Computer-verified
|-
| <math>\frac{\sqrt{33} \pm 1}{2\sqrt{3}}</math>
| 1/2
| <math>G_{420}</math>
| <math>0.48</math>
| Proposition 36
| Computer-verified
|}

== Bounds on <math>{\bf P}( \mathbf{c}(0) = \mathbf{c}(d_1) \mid \mathbf{c}(0) \neq \mathbf{c}(d_0) )</math> for 4-colourings ==

{| border=1
|-
! <math>d_1</math> !! <math>d_0</math> !! Lower bound !! Lower-bounding graph !! Upper bound !! Upper-bounding graph !! Notes
|-
| <math>1+(-1)^{1/3}</math>
| <math>2</math>
| 1/2
| H
|
|
| Equals <math>p_{\sqrt 3}/(1-p_2)</math>
|-
| <math>1+(-1)^{1/3}</math>
| <math>1+(-1)^{-1/3}</math>
|
|
| 1/2
| H
|
|}

== Proofs of bounds ==

One can compute some correlations of the coloring exactly:

=== Lemma 1 ===
Let <math>z,w \in {\bf C}</math> with <math>|z-w|=1</math>. Then
:<math> {\bf P}( \mathbf{c}(z) = c ) = \frac{1}{4}\quad (4)</math>
for all <math>c=1,\dots,4</math>,
:<math> {\bf P}( \mathbf{c}(z) = \mathbf{c}(w) ) = 0\quad (5),</math>
and
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c' ) = \frac{1}{12} \quad (6)</math>
for any distinct <math>c,c' \in \{1,2,3,4\}</math>. If <math>u</math> is at a unit distance from both z and w, then
:<math> {\bf P}( \mathbf{c}(z) = c; \mathbf{c}(w) = c'; \mathbf{c}(u) = c'' ) = \frac{1}{24} \quad (6')</math>

'''Proof''' By color invariance (2), the four probabilities in (4) are equal and sum to 1, giving (4). The claim (5) is immediate from (1). From (5) and color invariance, the 12 probabilities in (6) are equal and sum to 1, giving (6). The same argument gives (6').<math>\Box</math>

=== Lemma 2 ===
(Spindle argument) Let <math>|d| \geq 1/2</math>. Then
:<math> p_d \leq \frac{1}{2} \quad (7).</math>

'''Proof''' We can find an angle <math>\theta</math> with <math>|de^{i\theta}-d|=1</math>, then <math>\mathbf{c}(de^{i\theta}) \neq \mathbf{c}(d)</math> almost surely. This means that at least one of the events <math>\mathbf{c}(0) = \mathbf{c}(d)</math>, <math>\mathbf{c}(0) = \mathbf{c}(d e^{i\theta})</math> occurs with probability at most 1/2. The claim now follows from isometry invariance (3). <math>\Box</math>

=== Lemma 3 ===
(Using the K graph) We have
:<math>52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) + {\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} ) \geq 1 \quad (8).</math>

'''Proof''' Consider the 61-vertex graph <math>K</math> from [https://arxiv.org/abs/1804.02385 de Grey's paper]. It has 26 (isometric) copies of H, and thus 52 copies of the triangle <math>(1, e^{2\pi i/3}, e^{4\pi i/3})</math>. With probability at least <math>1 - 52 {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) </math>, none of these triangles are monochromatic. By the argument in that paper, this implies that the three linking diagonals <math>(-2, +2)</math>, <math>(-2 e^{2\pi i/3}, 2e^{2\pi i/3})</math>, <math>(-2 e^{4\pi i/3}, e^{-4\pi i/3})</math> are monochromatic. This gives the claim. <math>\Box</math>

=== Corollary 4 ===
(Existence of monochromatic <math>\sqrt{3}</math>-triangles) We have <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) \geq \frac{1}{104}</math>.

'''Proof''' The probability <math>{\bf P}( \mathbf{c}(-z) = \mathbf{c}(z) \hbox{ for } z = 2, 2e^{2\pi i/3}, 2e^{4\pi i/3} )</math> is at most <math>{\bf P}( \mathbf{c}(-2) = \mathbf{c}(2)) = p_4</math>, which by Lemma 2 is at most 1/2. The claim now follows from Lemma 3. <math>\Box</math>

=== Computer-verified Claim 5 ===
(Using the graph M) One has <math>{\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = 0</math> (Note this contradicts Corollary 4).

'''Proof''' This simply reflects the fact that there is no 4-coloring of the 1345-vertex graph M from [https://arxiv.org/abs/1804.02385 de Grey's paper] with its central copy of H containing a monochromatic triangle. One can use other graphs for this purpose, such as the 278-vertex graph <math>M_1</math> or <math>V \oplus V \oplus H</math>. <math>\Box</math>

=== Computer-verified Claim 6 ===
(Using the graph <math>V \oplus V \oplus H</math>) One has <math>p_{8/3} = 1</math> (note this contradicts Lemma 2).

'''Proof''' This reflects the fact that every 4-coloring of <math>V \oplus V \oplus H</math> must assign the same color to 0 and 8/3. There is also a 745-vertex subgraph of <math>V \oplus V \oplus H</math> with the same property. <math>\Box</math>

=== Lemma 7 ===
(Using <math>G_{40}</math>) We have
:<math>59 p_{\sqrt{11/3}} + p_{8/3} \geq 1</math>.

'''Proof''' This reflects the fact that every 4-coloring of the 40-vertex graph <math>G_{40}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which none of the 59 pairs of vertices at distance <math>\sqrt{11/3}</math> apart, will assign the same color to 0 and 8/3. (This is presumably human-verifiable.) <math>\Box</math>

=== Corollary 8 ===
One has
:<math> p_{\sqrt{11/3}} \geq \frac{1}{118}</math>.

'''Proof''' Combine Lemma 7 and Lemma 2. <math>\Box</math>

=== Lemma 9 ===
(Using <math>G_{49}</math>) One has
:<math>18 {\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq p_{\sqrt{11/3}} .</math>

'''Proof''' This reflects the fact that every 4-coloring of the 49-vertex graph <math>G_{49}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] in which 0 and <math>\sqrt{11/3}</math> have the same color, at least one of the 18 copies of <math>(1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3)</math> is monochromatic. This is potentially human-verifiable. <math>\Box</math>

=== Corollary 10 ===
(Existence of monochromatic <math>1/\sqrt{3}</math>-triangles) One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) \geq \frac{1}{2124}.</math>

'''Proof''' Combine Corollary 8 and Lemma 9. <math>\Box</math>

=== Computer-verified claim 11 ===
One has <math>{\bf P}( \mathbf{c}(1/3) = \mathbf{c}(e^{2\pi i/3}/3) = \mathbf{c}(e^{4\pi i/3}/3) ) = 0</math>. (This contradicts Corollary 10).

'''Proof''' This reflects the fact that the 627-vertex graph <math>G_{627}</math> from [https://arxiv.org/abs/1805.00157 Exoo-Ismailescu] does not have any 4-colorings with <math>1/3, e^{2\pi i/3}/3, e^{4\pi i/3}/3</math> monochromatic. <math>\Box</math>

=== Lemma 12 ===
For certain special distances d, one can improve the bound in Lemma 2:

If <math>k \geq 1</math> is a natural number, <math>j\in\mathbb{Z}</math>, <math>\gcd(j,2k+1)=1</math>, and <math>r = \frac{1}{2} \csc\left(\frac{j\pi}{2k+1}\right)</math> then
:<math> p_r \leq \frac{k}{2k+1},</math>
thus for instance
:<math> p_{\frac{1}{\sqrt{3}}} \leq \frac{1}{3}.</math>

'''Proof''' Observe that the regular 2k+1-polygon <math>r, re^{2\pi i/(2k+1)}, r e^{4\pi i/(2k+1)}, \dots, r^{4k\pi i/(k+1)}</math> has unit side lengths. By the pigeonhole principle, we conclude that at most k of these vertices can have the same color as the origin, and the claim follows. <math>\Box</math>

On the other hand, for <math>k=2,j=1</math> we also know from the regular pentagon of unit sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}+1}{2}} \leq \frac{2}{5} \quad (9)</math>
since any 4-coloring of that pentagon has either one or two monochromatic diagonals.

Similarly, for <math>k=2,j=2</math> we also know from the regular pentagon of <math>\frac{\sqrt{5}-1}{2}</math> sidelength that
:<math> \frac{1}{5} \leq p_{\frac{\sqrt{5}-1}{2}} \leq \frac{2}{5}</math>
since any 4-coloring of that pentagon has either one or two monochromatic edges. More generally, if <math>a,b,c,d,e</math> are the diagonal lengths of a pentagon with unit sides, then
:<math> 1 \leq p_a + p_b + p_c + p_d + p_e \leq 2.</math>

=== Lemma 13 ===
We have
:<math> 7 p_{\frac{1}{\sqrt{3}}} \geq p_{\sqrt{3}}</math>.

'''Proof''' Consider the unit rhombus <math>0, 1, e^{i\pi/3}, e^{-i\pi/3}</math> together with the centers <math>e^{i\pi/6}/\sqrt{3}, e^{-i\pi/6}/\sqrt{3}</math>. With probability <math>p_{\sqrt{3}}</math>, the two far vertices <math>e^{i\pi/3}, e^{-i\pi/3}</math> are the same color, and then 0,1 will be two other colors. This forces either one of the centers <math>e^{i\pi/6}/\sqrt{3}</math> of a triangle to have a common color with one of the vertices of that triangle, or the two centers must have the same color. Thus in any event one of the seven edges of distance <math>1/\sqrt{3}</math> is monochromatic, giving the claim. <math>\Box</math>

=== Corollary 14 ===
We have <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{728}</math>.

This slightly improves upon the lower bound of 1/2124 coming from Corollary 10.

'''Proof''' Combine Corollary 4 and Lemma 13. <math>\Box</math>

=== Lemma 15 ===
One has
:<math>p_{\sqrt{3}} + p_2 \geq \frac{2}{3}</math> and <math>2 p_{\sqrt{3}} + p_2 \geq 1</math>.

'''Proof''' As noted in de Grey's paper, there are essentially four 4-colorings of H. H has six edges of length <math>\sqrt{3}</math> and three of length <math>2</math>. If we let a denote the number of monochromatic <math>\sqrt{3}</math> edges and b the number of monochromatic <math>2</math>-edges, we see from inspection of all four colorings that <math>(a,b)</math> is either <math>(6, 0), (4,0), (2, 1)</math>, or <math>(0,3)</math>. In particular, one always has <math>\frac{a}{6} + \frac{b}{3} \geq \frac{2}{3}</math> and <math>2\frac{a}{6} + \frac{b}{3} \geq 1</math>. Taking expectations, we obtain the claim. <math>\Box</math>

=== Corollary 16 ===
We have <math>p_2 \geq \frac{1}{6}</math>, <math>p_{\sqrt{3}} \geq \frac{1}{4} </math> and <math>p_{\frac{1}{\sqrt{3}}} \geq \frac{1}{28}</math>.

'''Proof''' Combine Lemma 2, Lemma 15, and Lemma 13. <math>\Box</math>

=== Lemma 17 ===
If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths a,b,c. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also, thus
:<math>{\bf P}(\mathbf{c}(0) \neq \mathbf{c}(a)) + {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(b)) \geq {\bf P}(\mathbf{c}(0) \neq \mathbf{c}(c))</math>
and the claim follows. <math>\Box</math>

Note that Lemma 2 follows from the a=b, c=1 case of this lemma. Iterating this lemma starting with Lemma 2 we can also obtain slightly nontrivial upper bounds on <math>p_a</math> for small values of a, e.g. <math>p_a \leq 1 - 2^{-k}</math> when <math>a \geq 2^{-k}, k\in\mathbb{Z}^+</math>.

Further, we can generalise the a=b case to one in which the triangle is replaced by a (k+1)-gon of which one edge is 1 and the others are all equal, leading to the stronger result <math>p_a \leq 1 - 1/k</math> when <math>a \geq 1/k, k\in\mathbb{Z}^+ \land k>1</math>. Further strengthening is achieved by using <math>1/\sqrt{3}</math> as the long edge, given Lemma 12.

=== Lemma 18 ===
Whenever <math>d>0</math>, one has the inequalities
:<math> |p_{\phi d} - p_d| \leq \frac{2}{5}, p_{\phi d} + p_d \geq \frac{1}{5}, 2p_d - p_{\phi d} \leq 1, 2 p_{\phi d} - p_d \leq 1</math>
where <math>\phi := \frac{1+\sqrt{5}}{2}</math> is the golden ratio. Also we have
:<math> p_{d/\sqrt{3}} \leq \frac{1}{3} + p_d, \frac{1}{2} + \frac{1}{2} p_d.</math>

Note that this generalises (9), as well as a special case of Lemma 12.

'''Proof''' Consider the regular pentagon with sidelength <math>d</math>, so it also has 5 diagonals of length <math>\phi d</math>. Let <math>a \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic edges and let <math>b \in \{0,1,2,3,4,5\}</math> denote the number of monochromatic diagonals. Observe:
* <math>a,b</math> cannot both be zero (pigeonhole principle).
* <math>a</math> cannot be 4. Similarly, <math>b</math> cannot be 4.
* If <math>a=5</math>, then <math>b=5</math>, and conversely.
* If <math>a=0</math>, then <math>b=1,2</math>; similarly, if <math>b=0</math>, then <math>a=1,2</math>.

From this we observe the inequalities
:<math> |\frac{a}{5}-\frac{b}{5}| \leq \frac{2}{5}; \frac{a}{5} + \frac{b}{5} \geq \frac{1}{5}; 2 \frac{a}{5} - \frac{b}{5} \leq 1; 2\frac{b}{5} - \frac{a}{5} \leq 1</math>
and on taking expectations we obtain the first claim. Similarly, if one considers the colorings of an equilateral triangle of sidelength <math>d</math> together with its center, and counts the numbers <math>a,b \in \{0,1,2,3\}</math> of monochromatic edges of length <math>d</math> and <math>d/\sqrt{3}</math> respectively, one observes that one always has <math>\frac{b}{3} \leq \frac{1}{3} + \frac{2}{3} \frac{a}{3}, \frac{1}{2} + \frac{1}{2} \frac{a}{3}</math>, and on taking expectations one obtains the claim.<math>\Box</math>

The hexagon <math>H</math> has essentially four distinct colorings: the coloring <math>\hbox{2tri}</math> with two triangles, the coloring <math>\hbox{1tri}</math> with one triangle, the coloring <math>\hbox{axisym}</math> that is symmetric around an axis, and the coloring <math>\hbox{centralsym}</math> that is symmetric around the central point. This gives four probabilities <math>p_{H = 2tri}, p_{H = 1tri}, p_{H = axisym}, p_{H = centralsym}</math> that sum to 1. By counting the number of monochromatic edges of length <math>\sqrt{3}, 2</math> respectively, one also obtains the identities
:<math>p_{\sqrt{3}} = p_{H = 2tri} + \frac{2}{3} p_{H = 1tri} + \frac{1}{3} p_{H = axisym}; \quad p_2 = \frac{1}{3} p_{H=axisym} + p_{H=centralsym}</math>
which reproves Lemma 15. Also
:<math> {\bf P}( \mathbf{c}(0) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) ) = p_{H = 2tri} + \frac{1}{2} p_{H=1tri}.</math>
Any 4-coloring of L contains at least one triangle within one of its 52 copies of H, thus
:<math> p_{H = 2tri} + \frac{1}{2} p_{H=1tri} \geq \frac{1}{52}</math>
which reproves Corollary 4.

=== Lemma 19 ===
(Hubai) One has <math>p_{H = 1tri} + p_{H = axisym} \geq \frac{1}{10}</math>.

'''Proof''' Consider five copies of H centred at 0,1,2,3,4. With probability at least <math>1 - 5( p_{H = 1tri} + p_{H = axisym} )</math>, none of these copies of H are colored 1tri or axisym, and so must be colored 2tri or centralsym. One can check then that if one of the copies is colored 2tri, then so is any adjacent copy; thus all five copies are colored 2tri, or all five are colored centralsym. In either case we see that -1 and 5 are colored the same color. Comparing with Lemma 2 then gives the claim. <math>\Box</math>

=== Theorem 20 ===
One has <math>p_{H = 1tri} > 0</math>.

'''Proof''' Suppose for contradiction that <math>p_{H = 1tri} = 0</math>. One can then run a version of the de Bruijn-Erdos argument to obtain a coloring in which 1tri hexagons are completely nonexistent (since there are arbitrarily large finite colorings with this property). Consider the triangular lattice <math>{\bf Z}[e^{\pi i/3}]</math>. We 2-color the edges of this lattice by coloring an edge black if it is the short diagonal of a unit rhombus with monochromatic long diagonal, and white otherwise. The four colorings of hexagons lead to four possible colorings at each vertex:

* If H is colored 2tri, then all six edges to the centre of H are black.
* If H is colored 1tri, then two edges to the centre of H at 120 degree angles are white, the other four are black.
* If H is colored axisym, then two opposing edges of the centre of H are black, the other four are white.
* If H is colored centralsym, then all six edges to the centre of H are black.

In particular, as we are assuming no 1tri hexagons, the faces cut out by the black edges have angles 60 degrees, and thus must be equilateral triangles, sectors of angle 60, half-planes, or the entire plane. If there is at least one equilateral triangle, then the rest of the black edges must form an equilateral lattice with that triangle sidelength. This leads to only a small number of possible hexagon colorings in the lattice:

# Case 1: All edges white.
# Case 2: All edges black.
# Case 3.k: For some natural number <math>k \geq 2</math>, the length k edges joining adjacent vertices in some coset of <math>k \cdot {\mathbf Z}[ e^{\pi i/3} ]</math> are all black, and the remaining edges are white.
# Case 4: Each horizontal row consists either entirely of black edges, or entirely of white edges.
# Case 5: Each northwest row consists either entirely of black edges, or entirely of white edges.
# Case 6: Each northeast row consists either entirely of black edges, or entirely of white edges.
# Case 7: Six rays of black edges meeting at a common vertex; all other edges white.

Technically, Case 1 is contained in Cases 4,5,6 as written above, but this will not be an issue. One can view Case 7 as a limiting case <math>k=\infty</math> of Case 3.k; Case 2 is similarly the opposite limiting case <math>k=1</math>.

In the first case, the coloring is periodic with periods <math>2, 2 e^{\pi i/3}</math>. In the second case, it is periodic with periods <math>3, 3 e^{\pi i/3}</math>. In the third case, it is periodic with periods <math>3k, 3k e^{\pi i/3}</math>. Also note that for each k, one can check if Case 3.k holds by inspecting the coloring at a finite number of vertices. Thus the event that Case 3.k holds is "measurable" in the sense that a meaningful probability can be assigned. (But Cases 1,2,4,5,6 are not measurable events, they require an infinite number of points to be inspected, and the probability measure we are using is only finitely additive rather than infinitely additive.) In Case 4, the coloring is periodic with period 2; also, every coset of <math>2 \cdot {\bf Z}[e^{\pi i/3}]</math> is 2-colored. Similarly for Case 5 and 6 (where the periods are <math>2 e^{2\pi i/3}</math> and <math>2 e^{4\pi i/3}</math> respectively.)

Let <math>\alpha_k</math> be the probability that Case 3.k holds for the given value of <math>k</math>. Then <math> \sum_{k=2}^K \alpha_k \leq 1</math> for any k, hence <math>\sum_{k=2}^\infty \alpha_k \leq 1</math>. In particular, we can find <math>K_1</math> such that <math>\sum_{k={K_1}}^\infty \alpha_k \leq 0.1</math> (say). Let <math>P</math> be six times the least common multiple of <math>1,2,\dots,K_1</math>. Then the coloring is P- and <math>P e^{\pi i/3}</math>-periodic for Case 1, Case 2, and all Case 3.k with <math>k \leq K_1</math>. On the other hand, if <math>K_2</math> is sufficiently large depending on <math>P</math>, and Case 3.k holds for some <math>k \geq K_2</math>, then almost all of the hexagons are colored centralsym, which makes the coloring "almost <math>P, P e^{\pi i/3}</math>-periodic" in the sense that
:<math>{\bf c}(z+P e^{\pi i j/3}) = {\bf c}(z) \hbox{ for } j=0,1,2,3,4,5</math>
will hold for at least <math>0.9</math> of the lattice points <math>z \in {\bf Z}[e^{\pi i/3}]</math> with <math>|z| \leq K_2</math>. Similarly for Case 7 (which is sort of a <math>k=\infty</math> limiting case of Case 3.k.) Thus, with the probability <math> \geq 1 - \sum_{k=K_1}^{K_2} \alpha_k \geq 0.9</math>, the coloring of the seven vertices <math>{\bf c}(0), {\bf c}(P e^{\pi ij/3}, j=1,\dots,6</math> is (up to rotation and recoloring) one of the three patterns of the central and linking vertices in Figure 3 of Aubrey's paper, namely

* <math>{\bf c}(0) = {\bf c}(P) = {\bf c}(P e^{\pi i/3}) = {\bf c}(P e^{2\pi i/3}) = {\bf c}(P e^{3\pi i/3}) = {\bf c}(P e^{4\pi i/3}) = {\bf c}(P e^{5\pi i/3}) </math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{3\pi i/3})</math>; <math>{\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>;
* <math>{\bf c}(0) = {\bf c}(P') = {\bf c}(P' e^{\pi i/3}) = {\bf c}(P' e^{2\pi i/3}) = {\bf c}(P' e^{3\pi i/3})</math>; <math> {\bf c}(P' e^{4\pi i/3}) = {\bf c}(P' e^{5\pi i/3}) </math> for some <math>P' = P e^{\pi i j/3}</math>, <math>j=1,\dots,6</math>.

Using the spindling argument from Aubrey's paper, we conclude that the third possibility must in fact hold with probability at least 0.8; on the other hand, from Lemma 2 this scenario can only occur with probability at most 1/2, giving the required contradiction.
<math>\Box</math>

One should be able to refine this argument to show that <math>p_{H = 1tri} > c</math> for an absolute constant <math> c>0 </math>.

=== Lemma 21 ===
Providing a tighter bound for Lemma 17 with a more thorough proof: If <math>a,b,c > 0</math> are the lengths of a triangle, then <math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math>.

'''Proof''' Consider a triangle of side lengths <math>a,b,c</math>. Let <math>\left|z_2\right|=b,\left|a-z_2\right|=c</math>. If the c side is not monochromatic, then at least one of the other two sides must fail to be monochromatic also: <math>\mathbf{c}(a)\neq\mathbf{c}(z_2)\Rightarrow[\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)]</math>. <math>[A\Rightarrow B]\Rightarrow {\bf P}(A)\leq{\bf P}(B)</math> thus
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) \geq {\bf P}(\mathbf{c}(a) \neq \mathbf{c}(z_2)) = 1-p_c</math>
<math>{\bf P}(A\lor B) +{\bf P}(A\land B)={\bf P}(A)+{\bf P}(B)</math>, so
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)) + {\bf P}(\mathbf{c}(0)\neq\mathbf{c}(z_2)) - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>{\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\lor\mathbf{c}(0)\neq\mathbf{c}(z_2)) = 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
:<math>1-p_c \leq 2 - p_a - p_b - {\bf P}(\mathbf{c}(a)\neq\mathbf{c}(0)\neq\mathbf{c}(z_2))</math>
Using the law of cosines: <math>z_2=b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)</math>
:<math>{\bf P}\left(\mathbf{c}(a)\neq \mathbf{c}(0)\neq \mathbf{c}\left(b\exp\left(i\arccos\left(\frac{a^b+b^2-c^2}{2ab}\right)\right)\right) \right) + p_a + p_b \leq 1 + p_c</math> <math>\Box</math>

=== Lemma 22 ===

One has <math>3 p_{1/\sqrt{3}} \geq {\bf P}( \mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) )</math>.

'''Proof''' Let <math>z</math> be a complex number of magnitude <math>1/\sqrt{3}</math> that is a unit distance from 1. If <math>\mathbf{c}(1) = \mathbf{c}(e^{2\pi i/3}) = \mathbf{c}(e^{4\pi i/3}) = c</math> (say), then <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> cannot be colored with <math>c</math>; also, <math>z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> are the vertices of a unit equilateral triangle and thus must take on three different colors. By the pigeonhole principle, one of <math>0, z, e^{2\pi i/3} z, e^{4\pi i/3} z</math> must then take the same color as the origin, and the claim follows. <math>\Box</math>

=== Lemma 23 ===

One has <math>4 p_{(\sqrt{6} \pm \sqrt{2})/2} + p_{\sqrt{2}} \geq 1</math>. In particular, by Lemma 2, <math>p_{(\sqrt{6}+\sqrt{2})/2} \geq 1/8</math>.

'''Proof''' [ExIs2018b] We just prove the claim for the + sign (the - sign can then be obtained after applying the Galois conjugacy that maps <math>\sqrt{3}</math> to <math>-\sqrt{3}</math>, leaving <math>\sqrt{2}</math> unchanged). Set <math>d := \frac{\sqrt{6}+\sqrt{2}}{2}</math>, and consider the five vertices

:<math>0, e^{5\pi i/4}, e^{5\pi i/4} + d, e^{5\pi i/4} + e^{\pi i/3} d, e^{5\pi i/4} + (e^{\pi i/3}-i)d.</math>

One can check that of the ten edges determined by these five vertices, five have unit length, four have length d, and the remaining distance (from 0 to <math>e^{5\pi i/4}+d</math>) has distance <math>\sqrt{2}</math>. Since <math>\mathbf{c}</math> must make one of the latter five edges monochromatic, the claim follows. <math>\Box</math>

=== Lemma 24 ===

One has <math>p_{\sqrt{2}} \geq \frac{1}{14}</math>.

'''Proof''' In page 7 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 20 unit distance edges and 14 edges of length <math>\sqrt{2}</math> is constructed. The coloring <math>\mathbf{c}</math> must make one of the 14 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 25 ===

Let <math>d = \frac{1}{2} \sqrt{3^{1/4} \cdot 2 \sqrt{2} + 2 \sqrt{3} + 2}</math> and <math>e = \frac{3^{1/4} \sqrt{2} + \sqrt{3} - 1}{2}</math>.
Then one has <math>14 p_d + p_e \geq 1</math>. In particular, by Lemma 2, <math>p_d \geq 1/28</math>.

'''Proof''' In page 9 of [ExIs2018b], a non-4-colorable graph of 13 vertices with 19 unit edges, 14 edges of length d, and one edge of length e is constructed. The coloring <math>\mathbf{c}</math> must make one of the 15 latter edges monochromatic, and the claim follows. <math>\Box</math>

=== Lemma 26 ===

Let <math>d = \sqrt{3/2 + \sqrt{33}/6}</math>. Then <math>7 p_d \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_d \geq \frac{1}{196}</math>.

'''Proof''' In page 11 of [ExIs2018b], a graph of nine vertices consisting of 12 unit edges and 7 edges of length d is constructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Thus, <math>\mathbf{c}</math> can only make the AB edge monochromatic if one of the seven length d edges is monochromatic. The claim follows.
<math>\Box</math>

=== Lemma 27 ===

One has <math>27 p_{\sqrt{5/3}} \geq p_{1/\sqrt{3}}</math>. In particular, by Corollary 16, <math>p_{\sqrt{5/3}} \geq \frac{1}{756}</math>.

'''Proof''' In page 13 of [ExIs2018], a graph of 33 vertices with some unit edges and 27 edges of length <math>\sqrt{5/3}</math> is contructed with the property that any 4-coloring of this graph cannot have two specific vertices A,B (which are distance <math>1/\sqrt{3}</math> apart) monochromatic. Now repeat the proof of Lemma 26. <math>\Box</math>

=== Computer-verified claim 28 ===

One has <math>p_{2/\sqrt{3}} \geq \frac{1}{177}</math>.

'''Proof''' In page 15 of [ExIs2018], a 5-chromatic graph of 103 vertices, 312 unit edges, and 177 edges of length <math>2/\sqrt{3}</math> is constructed. <math>\mathbf{c}</math> must make one of the latter edges monochromatic, giving the claim. <math>\Box</math>

=== Lemma 29 ===

One has <math>p_{(\sqrt{6} \pm \sqrt{2})/2} \geq 1/6</math> (this improves the bound in Lemma 23).

'''Proof''' Use graphs 505 and 507 from [S2004] and the spindle bound. <math>\Box</math>

=== Lemma 30 ===
For <math>m > n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' <math>n</math> colors and <math>m</math> points necessitates at least 2 having equal color. I.e.

:<math>{\bf P}\left(\bigvee_{k=0}^n \bigvee_{j=k+1}^n\ \mathbf{c}(z_k) = \mathbf{c}(z_j)\right) = 1</math>

The lemma then follows immediately from the fact:
:<math>{\bf P}\left(\bigcup_{k} E_k\right) \leq \sum_{k} {\bf P}\left(E_k\right) \,\Box</math>

=== Corollary 31 ===
Let <math>\lvert z_k\rvert=1</math>. Then for <math>m \geq n</math> and an <math>n</math>-coloring of the complex plane, <math>\sum_{k=1}^m\sum_{j=k+1}^m{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(z_j) \right) \geq 1</math>.

'''Proof''' Use lemma 30 on the set <math>\left\{z_k \bigg\vert 1\leq k\leq m \land k\in\mathbb{Z}\right\}\cup\{0\}</math>. Simplify using <math>{\bf P}\left(\mathbf{c}(z_k) = \mathbf{c}(0) \right)=0</math>.
<math>\Box</math>

=== Lemma 32 ===
For <math>x\in\mathbb{R}</math> and an <math>n</math>-coloring of the plane, <math>\sum_{k=1}^{n-1}\left(n-k\right){\bf P}\left(\mathbf{c}\left(0\right) = \mathbf{c}\left( 2\sin\left(\frac{kx}{2}\right) \right) \right) \geq 1</math>.

'''Proof''' Use corollary 31 on the set <math>\left\{e^{ikx} \bigg\vert 0\leq k < n \land k\in\mathbb{Z}\right\}</math>. and simplify by grouping lengths.
<math>\Box</math>

=== Corollary 33 ===
Interesting(easy to simplify results of) values for <math>x</math> in Lemma 32 are in <math>\left\{x \bigg\vert \sin\left(\frac{kx}{2}\right)=1 \land 1\leq k < n \land k\in\mathbb{Z}\right\}</math>.
For 4-colorings, this gives
:<math>2p_{\sqrt 3}+p_2 \geq 1</math>
:<math>3p_{(\sqrt 3-1)/\sqrt 2}+p_{\sqrt 2} \geq 1</math>
:<math>3p_{2\sin(\pi/18)}+2p_{2\sin(\pi/9)} \geq 1</math>

=== Lemma 34 ===
Generalizing the note of Lemma 17, <math>\lvert d_1\rvert= d_1 > \lvert d_0\rvert= d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>

'''Proof''' let <math>\lvert z_{j+1} -z_j\rvert=d_0 > 0, \lvert z_{j+n} -z_0\rvert=d_1>0</math>.

Base case, <math>n=2</math>, by Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle:
:<math>2d_0\geq d_1\Rightarrow 2p_{d_0}\leq 1+p_{d_1}</math>
The inductive step is Lemma 17 using the <math>\left(z_n-z_0,z_n-z_{n-1},z_{n-1}-z_0\right)</math> triangle. After induction:
:<math>[n\geq 2\land nd_0\geq d_1]\Rightarrow np_{d_0}\leq n-1+p_{d_1}</math>
Substitute <math>n=\left\lceil\frac{d_1}{d_0}\right\rceil</math>, simplify, rearrange:
:<math>d_1 > d_0\Rightarrow (1-p_{d_1})\leq \left\lceil\frac{d_1}{d_0}\right\rceil(1-p_{d_0})</math>
<math>\Box</math>

=== Proposition 35 ===

If <math>d > 1/\sqrt{2}</math> obeys the inequalities
:<math>\sin( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
and
:<math>\cos( 2 \mathrm{arcsin} \frac{1}{2d} ) \geq \frac{1}{4d}</math>
then
:<math>p_d \leq \frac{1}{2} - \frac{1}{188}.</math>

(One can check that the conditions are obeyed precisely when <math>d \geq \frac{\sqrt{33}-1}{8} = 0.84307\dots</math>.)

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. Since <math>AB</math> is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the triangle <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>.

Now let <math>BADE</math> be a rhombus with sidelengths d and <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. By the hypotheses, the diagonals BD, AE of this rhombus have length at least 1/2, and hence are monochromatic with probability at most 1/2 by Lemma 2. As above, ABD and BDE are each monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>. As BD is monochromatic with probability at most <math>1/2</math>, we conclude that BADE is monochormatic with probability at least <math>\frac{1}{2}-6\delta</math>.

Now let <math>EDFG</math> be another rhombus congruent to <math>BADE</math>. As BD, AE have length at least 1/2, at least one of the long diagonals BF, AG have length at least 1/2 (the diagonal opposite an obtuse or right-angled triangle will work). Let's say BF has length at least 1/2. As BADE and EDFG are both monochromatic with probability at least <math>\frac{1}{2}-6\delta</math>, and the common edge DE is monochromatic with probability <math>\frac{1}{2}-\delta</math>, we conclude that the entire configuration ABDEFG is monochromatic with probability at least <math>\frac{1}{2}-11\delta</math>. In particular the pentagon ABDEF is monochromatic with at least this probability. However, in this pentagon, the five edges BA, AD, DE, EB, EF are monochromatic with probability at most <math>\frac{1}{2}-\delta</math>, and the other five edges are monochromatic with probability at most <math>\frac{1}{2}</math> by Lemma 2. Thus the probability that at least one of the edges of this pentagon is monochromatic is at most <math>(\frac{1}{2}-11\delta) + 5 \times 10\delta + 5 \times 11\delta = \frac{1}{2}+94\delta</math>. On the other hand, by the pigeonhole principle, this probability is 1. The claim follows. <math>\Box</math>

=== Proposition 36 ===

If <math>d \ge \frac{2}{\sqrt{15}} = 0.5163\dots</math>, then <math>p_d \leq \frac{1}{2} - \frac{1}{62}</math>,

if <math>d \ge \frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math>, then <math>p_d \leq 0.48</math>,

and <math>\limsup_d p_d\leq \frac{323}{675}=0.4785\ldots</math> (so <math>p_d<0.4786</math> if <math>d</math> is large enough).

'''Proof''' Write <math>p_d = \frac{1}{2}-\delta</math>. Let <math>ABC</math> be any isosceles triangle of sidelengths <math>|AB|=|AC|=d, |BC|=1</math>; this subtends an acute angle of <math>2\mathrm{arcsin} \frac{1}{2d}</math> at A. Then AB and AC are each individually monochromatic with probability <math>\frac{1}{2}-\delta</math>, but cannot both be monochromatic. Thus, the probability that <math>AB</math>, <math>AC</math> are both not monochromatic is at most <math>2\delta</math>.

Now let <math>ABCD</math> be a quadrilateral with <math>|AB|=|AC|=|AD|=d, |BC|=|CD|=1</math>, so <math>\angle BAD = 4 \mathrm{arcsin} \frac{1}{2d}</math>. A simple calculation shows that if <math>d \ge \frac{2}{\sqrt{15}}</math>, then <math>|BD| \ge \frac{1}{2}</math>.
If <math>AB</math> is monochromatic, then <math>AC</math> is not monochromatic, and thus by the previous paragraph <math>AD</math> is monochromatic outside of an event of probability at most <math>2\delta</math>. By inclusion-exclusion, we conclude that outside of the event that <math>AB</math> is monochromatic, the probability that <math>AD</math> is monochromatic is at most <math>2\delta</math>.

Now let <math>ADE</math> the <math>180^\circ</math> rotation of <math>ADB</math> around the midpoint of <math>AD</math>, and let <math>FDE</math> be the <math>180^\circ</math> rotation of <math>ADE</math> around the midpoint of <math>DE</math>. By the hypotheses, the line segments <math>AE, BD, BE, BF, DF</math> all have length at least 1/2. Let <math>{\mathcal E}</math> be the event that at least one of <math>AB, AD, DE, EF</math> is monochromatic. By the previous paragraph, this event occurs with probability at most <math>\frac{1}{2}-\delta+2\delta+2\delta+2\delta = \frac{1}{2}+5\delta</math>.

By previous considerations, <math>ABD</math> is monochromatic with probability at least <math>\frac{1}{2}-3\delta</math>, and this event lies in <math>{\mathcal E}</math>. On the other hand, <math>BD</math> is monochromatic with probability at most 1/2 by Lemma 2. We conclude that outside of <math>{\mathcal E}</math>, <math>BD</math> is only monochromatic with probability at most <math>3\delta</math>. A similar argument (replacing <math>ABD</math> by <math>DAE</math> or <math>EDF</math>) shows that outside of <math>{\mathcal E}</math>, <math>AE</math> is monochromatic with probability at most <math>3\delta</math>, and similarly for <math>DF</math>. Now we consider <math>EB</math>. By previous considerations, the probability that <math>ABDE</math> is monochromatic is at least <math>\frac{1}{2}-5\delta</math>, and this event lies inside <math>{\mathcal E}</math>. Thus, outside of <math>{\mathcal E}</math>, the probability that <math>EB</math> is monochromatic is at most <math>5\delta</math>; similarly for <math>AF</math>. Finally, the probability that <math>BF</math> is monochromatic outside of <math>{\mathcal E}</math> is at most <math>7\delta</math>. We conclude that outside of an event of probability
:<math>\frac{1}{2}+5\delta + 3\delta+3\delta+3\delta+5\delta+5\delta+7\delta = \frac{1}{2} + 31\delta,</math>
none of the ten edges connecting <math>A,B,D,E,F</math> are monochromatic. But by the pigeonhole principle, this cannot occur in a 4-coloring, hence <math>\frac{1}{2} + 31 \delta \geq 1</math>, and the first claim follows.

For the second claim, we need to use an iterative argument, by feeding the bounds obtained back into the place in the proof where Lemma 2 is currently invoked. To have all occurring distances stay larger than <math>d</math>, we only need to check <math>|BD| \ge d</math>. Equality occurs when <math>ABD</math> is an equilateral triangle, which means that <math>ABC</math> and <math>ACD</math> are isosceles triangles with sides <math>d,d,1</math> and either with angles <math>150^\circ,15^\circ,15^\circ</math>, or with angles <math>30^\circ,75^\circ,75^\circ</math>. From here calculation gives <math>d \ge \frac{1}{2sin(75^\circ)}=\frac{\sqrt{3}-1}{\sqrt{2}}=0.5176\dots</math> and <math>d \le \frac{1}{2sin(15^\circ)}=\frac{\sqrt{3}+1}{\sqrt{2}}=1.9318\dots</math>, but the upper bound is not really important, as for us it is enough that <math>|BD|</math> always stay above <math>d_0=\frac{\sqrt{3}-1}{\sqrt{2}}</math>, which occurs everywhere above this value. Now pick a <math>d\ge d_0</math> for which <math>p_d\ge \frac{1}{2}-\delta-\varepsilon</math>, where <math>\sup_{d\ge d_0} p_d= \frac{1}{2}-\delta</math> and <math>\varepsilon</math> is a small positive number. The calculation of the first case gives <math>\frac{1}{2}+5\delta + 2\delta+2\delta+2\delta+4\delta+4\delta+6\delta+O(\varepsilon) =\frac{1}{2} + 25 \delta+O(\varepsilon) \geq 1</math>, from which <math>\delta\ge 0.02</math> if we choose <math>\varepsilon</math> small enough.

To prove the last claim, we modify the construction; we obtain <math>F</math> by reflecting <math>B</math> to <math>AD</math>, to win <math>2\delta</math> in the last step of the calculation. To invoke Lemma 2, we need (among other things) that <math>EF</math> is at least 1/2, and to iterate in a straight-forward way, we would need a value <math>d_0</math> for which <math>EF</math> is at least <math>d_0</math> if <math>AB</math> is <math>d\ge d_0</math>, but such a <math>d_0</math> doesn't exist. We can, however, still iterate in a weaker sense, as <math>6</math> of the occurring <math>10</math> distances tend to infinity as <math>d=|AB|</math> tends to infinity, and the remaining <math>4</math> are also larger than <math>\frac{\sqrt{3}-1}{\sqrt{2}}</math>, so their probability of them being monochromatic is at most <math>0.48=(0.5-\delta)+(\delta-0.02)</math>. What we get eventually is <math>\frac{1}{2} + 25 \delta-2\delta+ 4(\delta-0.02)+O(\varepsilon) =0.42 + 27 \delta+O(\varepsilon) \geq 1</math>, from which <math>p_d\le \frac{323}{675}=0.4785\ldots</math> for <math>d</math> large enough.
<math>\Box</math>

=== Lemma 37 ===

One has <math>\sup_{0 < d < 2} p_d \geq 1/3</math>.

'''Proof''' For a large integer <math>n</math>, consider the points <math>e^{2\pi i j/n}</math> with <math>j=1,\dots,n</math>. Any unit distance coloring will color these points in at most 3 colors, hence divides the n points into three color classes of some size <math>n_1,n_2,n_3</math>. The number of monochromatic pairs is then
:<math>\frac{n_1(n_1-1)}{2} + \frac{n_2(n_2-1)}{2} + \frac{n_3(n_3-1)}{2} = \frac{1}{2} (n_1^2+n_2^2+n_3^2) + O(n) \geq \frac{1}{6} n^2 + O(n)</math>
by Cauchy-Schwarz. Thus at least <math>1/3-O(1/n)</math> of the pairs are monochromatic. Taking expectations and using the pigeonhole principle, we conclude that one of the distances has a probability at least <math>1/3 -O(1/n)</math> of being monochromatic, giving the claim. <math>\Box</math>

=== Lemma 38 ===

Let ABC be a unit-edge equilateral triangle, and let D be an arbitrary point. Let <math>|AD|, |BD|</math> and <math>|CD|</math> be <math>x,y,z</math> respectively. Then <math>p(x)+p(y)+p(z) \leq 1</math>. In particular, examining the case e=f, if <math>p(d) \geq k</math> then <math>p(\sqrt(d(d \pm \sqrt 3) + 1) \leq (1-k)/2</math>.

'''Proof''' At most one of <math>AD, BD</math> and <math>CD</math> can be monochromatic. <math>\Box</math>

Note: A consequence is that a 4-chromatic unit-distance graph G can demonstrate CNP <math>> 4</math> if, for the {x,y,z} arising from some choice of D above, G contains three equal-sized non-empty sets v_x, v_y, v_z of vertex-pairs such that (a) each vertex-pair within v_x is at distance x (resp. y and z), and (b) in any 4-colouring of G, more than 1/3 of the vertex-pairs in the union of the three sets are monochromatic. Note that this demonstration does not require that v_x contain all the vertex-pairs of G that are at distance x (resp. y and z), nor even that the graph {A,B,C,D} which gives rise to {x,y,z} be a subgraph of G. It seems plausible to find such a graph that is small (and/or symmetrical) enough that its colourings can be human-analysed to establish this property.

== Simplification rules for triplets of points in the complex plane ==
Deduced from the rule <math>{\bf P}(A\land B)+{\bf P}(A\land \lnot B)={\bf P}(A)</math>.
:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) + {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) = {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) = {\mathbf c}(z_2) ) - {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) ) = {\bf P}( {\mathbf c}(z_0) = {\mathbf c}(z_1) ) - {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

:<math>{\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) \neq {\mathbf c}(z_0) ) + {\bf P}( {\mathbf c}(z_1) \neq {\mathbf c}(z_2) = {\mathbf c}(z_0) ) = {\bf P}( {\mathbf c}(z_0) \neq {\mathbf c}(z_1) \neq {\mathbf c}(z_2) )</math>

== Proofs of bounds for conditional probabilities ==
The trivial case, valid where <math>\left|d\right|\neq 1</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) = {\mathbf c}(d) )=0</math>

Trivial plus Baye's Theorem, valid where <math>d\neq 0</math>:
:<math>x\in\mathbb{R}\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) )=\frac{{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )}\leq 1</math>
Rearrange:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{ix}) )+{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1</math>

Spindle method: for <math>\left|d\right|=d\geq 1/2</math> and <math>\theta=2\text{arcsin}\left(\frac{1}{2d}\right)</math>:
:<math>{\bf P}( {\mathbf c}(0) = {\mathbf c}(d+e^{i\theta}) \mid {\mathbf c}(0) \neq {\mathbf c}(d) ) = \frac{1}{1-{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )} - 1\leq 1</math>
which is another way to see <math>\left|d\right|=d\geq 1/2\Rightarrow{\bf P}( {\mathbf c}(0) = {\mathbf c}(d) )\leq 1/2</math>.

== Further questions ==
* For <math>n,m\geq CNP</math>, what consistent relationships exist between <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert n\text{ colors}\right)</math> and <math>{\bf P}\left(\mathbf{c}(0)=\mathbf{c}(d)\bigg\vert m\text{ colors}\right)</math>? How can these relationships be used to sharpen arguments of the probabilistic formulation?

[[Category:Polymath16]]