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.

Notable unit distance graphs

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
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
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	Contains three Moser spindles
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 \sqrt(3) 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
N	20425	151311	Contains 52 copies of M arranged around the H-copies of L	Not 4-colorable
G	1581	7877	N "shrunk" by stepwise deletions and replacements of vertices	Not 4-colorable

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.
• Polymath proposal: finding simpler unit distance graphs of chromatic number 5, Aubrey de Grey, 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.
• https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/, Dustin Mixon, Apr 13 2018.

Code and data

• 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

Wikipedia

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

Bibliography

• [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.