Hadwiger-Nelson problem

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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.
    • Improve the corresponding bound in higher dimensions.
    • Improve the current record of 105/29 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

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 roots of unity [math]\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))[/math] for various natural numbers t.

Name Number of vertices Number of edges Structure Group Colorings
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
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
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
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
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
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
[math]L_1[/math] 97 Has 40 copies of H All 4-colorings contain an H with a monochromatic [math]\sqrt{3}[/math]-triangle
[math]L_2[/math] 120 354 All 4-colorings contain an H with a monochromatic [math]\sqrt{3}[/math]-triangle
T 9 15 Contains one Moser spindle and useful symmetry; three vertices form an equilateral triangle
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]
V 31 30 Unit vectors at angles consistent with three interlocking Moser spindles
[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]V_a[/math] 25 24 Star graph
[math]V_b[/math] 25 24 Star graph
[math]V_x[/math] 13 12 Subgraph of [math]V_a[/math]
[math]V_z[/math] Subgraph of [math]V_a[/math]; shares a line of symmetry with [math]V_a[/math]
[math]V_y[/math] 13 12 Subgraph of [math]V_b[/math]
[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]
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])
[math]W_1[/math] Trimmed product of V with itself ([math]\mathrm{trim}(V \oplus V, 1.95)[/math])
M 1345 8268 Cartesian product of W and H ([math]W \oplus H[/math]) [math]{\bf Z}[\omega_1, \omega_3][/math] All 4-colorings have a monochromatic triangle in the central copy of H
[math]M_1[/math] 278 Deleting vertices from M while maintaining its restriction on H All 4-colorings have a monochromatic triangle in the central copy of H
[math]M_2[/math] 7075 Sum of H with a trimmed product of [math]V_1[/math] with itself Not 4-colorable
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
[math]G_0[/math] 1585 7909 N "shrunk" by stepwise deletions and replacements of vertices Not 4-colorable
G 1581 7877 Deleting 4 vertices from [math]G_0[/math] Not 4-colorable
[math]G_1[/math] 1577 Deleting 8 vertices from [math]G_0[/math] Not 4-colorable
[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
[math]G_3[/math] 826 4273 Not 4-colorable
[math]G_4[/math] 803 4144 Not 4-colorable
R Union of [math]W_1[/math] and a rotated copy of [math]W_1[/math]
[math]\mathrm{trim}(R \oplus H, 1.67)[/math] 2563 Trimmed sum of R and H Not 4-colorable
[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
No name assigned 745 Subgraph of [math]V \oplus V \oplus H[/math] Has two vertices forced to be the same color in a 4-coloring
[math]V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H[/math] 3085 Not 4-colorable
[math]V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H[/math] 3049 Not 4-colorable
No name assigned 1951 Trimmed version of [math]V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H[/math] Not 4-colorable
[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
[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
[math]G_{79}[/math] 79 165 Spindling of [math]G_{40}[/math] Any 4-coloring has a monochromatic [math]\sqrt{11/3}[/math]-edge
[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
[math]G_{51}[/math] 51 Has a specific [math]1/\sqrt{3}[/math]-triangle which cannot be monochromatic in a 4-coloring of plane
[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
No name assigned 103 All 4-colorings contain a monochromatic [math]2/\sqrt{3}[/math]-edge

Lower bounds

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 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.

Every unit distance graph with at most 16 vertices is 5-colorable.

A "tile-based" colouring of the plane must have at least 6 colours, as shown by Townsend [Tow2005]; the same proof with a minor error was also derived by Woodall [W1973]. In [T1999], Thomassen showed that any tiling-based 6-coloring would have to be 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).

Virtual edge

Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion.

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:

Probabilistic formulation

See Probabilistic formulation of Hadwiger-Nelson problem.

Further questions

  • What are the independence ratios of the above unit distance graphs?
  • What are the fractional chromatic numbers of these graphs?
  • What are the Lovasz numbers of these graphs?
  • 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. Varga and 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 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 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 this). David Speyer suggests looking at [math]\mathbb{Z}[\frac{1+\sqrt{-71}}{2}][/math] next.

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