The Hadwiger–Nelson Problem

Find the chromatic number \(\chi\) of the plane: colour every point of \(\mathbb{R}^2\) so that any two points at Euclidean distance exactly \(1\) receive different colours. Known: \(5 \le \chi \le 7\).

Posed around 1950 (Nelson attributed; Hadwiger published related results 1945/1961). Lower bound 4 came with the Moser spindle (1961); upper bound 7 from hexagonal tilings. In April 2018 biologist Aubrey de Grey — Polymath8 veteran of another kind — exhibited a 1,581-vertex unit-distance graph forcing five colours, igniting Polymath16. Note the family resemblance to our AI-resolved unit-distance entry: this is the colouring sibling of Erdős's packing question.