Hadwiger-Nelson problem

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]\displaystyle{ 4 \leq CNP \leq 7 }[/math] are classical; recently [deG2018] it was shown that [math]\displaystyle{ 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

• Polymath proposal: finding simpler unit distance graphs of chromatic number 5, Aubrey de Grey, Apr 10 2018. (Active discussion thread)
• Polymath16, first thread: Simplifying de Grey’s graph, Dustin Mixon, Apr 14, 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]\displaystyle{ CNP }[/math]; in particular, if one can find a unit distance graph with no 4-colorings, then [math]\displaystyle{ CNP \geq 5 }[/math].

Name	Number of vertices	Number of edges	Structure	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	Not 3-colorable
Golomb graph	10	18	Contains the center and vertices of a hexagon and equilateral triangle	Not 3-colorable
H	7	12	Vertices and center of a hexagon	Has essentially four 4-colorings, two of which contain a monochromatic triangle
J	31	72	Contains 13 copies of H	Has essentially six 4-colorings in which no H has a monochromatic triangle
K	61	150	Contains 2 copies of J	In all 4-colorings lacking an H with a monochromatic 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 triangle
[math]\displaystyle{ L_1 }[/math]	97		Has 40 copies of H	All 4-colorings contain an H with a monochromatic triangle
[math]\displaystyle{ L_2 }[/math]	120	354		All 4-colorings contain an H with a monochromatic 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
V	31	30	Unit vectors at angles consistent with three interlocking Moser spindles
W	301	1230	Cartesian product of V with itself, minus vertices at more than [math]\displaystyle{ \sqrt{3} }[/math] from the centre
M	1345	8268	Cartesian product of W and H	All 4-colorings have a monochromatic triangle in the central copy of H
[math]\displaystyle{ 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
N	20425	151311	Contains 52 copies of M arranged around the H-copies of L	Not 4-colorable
[math]\displaystyle{ 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]\displaystyle{ G_0 }[/math]	Not 4-colorable
[math]\displaystyle{ G_1 }[/math]	1577		Deleting 8 vertices from [math]\displaystyle{ G_0 }[/math]	Not 4-colorable
[math]\displaystyle{ G_2 }[/math]	874	4461	Juxtaposing two copies of M and shrinking	Not 4-colorable

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?
  • The Lovasz theta function value of Lovasz number of [math]\displaystyle{ G_2 }[/math] at the complement is in the interval [3.3746, 3.3748].
• What about the Erdos unit distance graph ([math]\displaystyle{ n }[/math] vertices, [math]\displaystyle{ n^{1+c/\log\log n} }[/math] edges)?

Blog posts and other online forums

• Has there been a computer search for a 5-chromatic unit distance graph?, Juno, Apr 16, 2016.
• The chromatic number of the plane is at least 5, Jordan Ellenberg, Apr 9 2018.
• Aubrey de Grey: The chromatic number of the plane is at least 5, Gil Kalai, Apr 10 2018.
• The chromatic number of the plane is at least 5, Dustin Mixon, Apr 10, 2018.
• Amazing progress on long-standing problems, Scott Aaronson, Apr 11 2018.
• The chromatic number of the plane is at least 5, Part II, Dustin Mixon, Apr 13 2018.

Code and data

This dropbox folder will contain most of the data and images for the project.

Some specific files:

• The 1585-vertex graph in DIMACS format
• A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format
• The vertices of this graph in explicit Sage notation
• The graph [math]\displaystyle{ G_2 }[/math]: vertices (Mathematica) edges (DIMACS)
• The densest unit-distance graphs on an [math]\displaystyle{ n\times n }[/math] grid for [math]\displaystyle{ n=10,20,\ldots,100 }[/math] (DIMACS format).

Code:

• MATLAB script for computing Lovasz number

Software:

• CVX
• Glucose 4.0
• Minisat

Wikipedia

• Chromatic number
• de Bruijn-Erdos theorem
• Hadwiger-Nelson problem
• Lovasz number
• Moser spindle

Bibliography

• [CR2015] D. Cranston, L. Rabern, The fractional chromatic number of the plane, arXiv:1501.01647
• [deBE1951] N. G. de Bruijn, P. Erdős, 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, The chromatic number of the plane is at least 5, arXiv:1804.02385
• [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
• [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
• [P1998] D. Pritikin, All unit-distance graphs of order 6197 are 6-colorable, Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
• [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.