Hadwiger-Nelson Problem
Determine how many colors are needed to color the plane so points distance 1 apart differ.
The chromatic number of the plane is known to be at least 5 and at most 7. The problem is easy to draw as a unit-distance graph but remains open after decades.
Formula
The minimum number of colors needed so no two points at distance 1 share a color lies between 5 and 7.
Aubrey de Grey constructed a unit-distance graph on 1,581 vertices requiring 5 colors, breaking the 1950s bound of 4.
Coloring the plane with regular hexagons of diameter slightly less than 1 gives a valid 7-coloring — unimproved since 1950.
Summary
The chromatic number of the plane is known to be at least 5 and at most 7. The problem is easy to draw as a unit-distance graph but remains open after decades.
