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].

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

• Hadwiger-Nelson problem
• Lovasz number
• Moser spindle

Bibliography

• [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, PortugaliaeMath. 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.