The chromatic number of the plane (Hadwiger–Nelson problem)
Color every point of the plane so that any two points at distance exactly 1 receive different colors. What is the smallest number of colors for which this is possible? It is known that the answer is 5, 6 or 7. Determine it.
History and references
Still open as of 2026.
Edward Nelson formulated the question in 1950 and Martin Gardner published it in 1960, while Hugo Hadwiger had studied a closely related covering problem in 1945. A tiling by regular hexagons of diameter slightly less than 1, colored in a repeating pattern, uses 7 colors. The Moser spindle, a unit distance graph with 7 vertices, shows that 3 colors are not enough. For decades the bounds stayed at 4 and 7. In 2018 Aubrey de Grey found a unit distance graph with 1581 vertices that cannot be colored with 4 colors, so at least 5 are needed, and a Polymath project later found much smaller examples. The answer may even depend on which axioms of set theory are assumed.
References
Solutions 0
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