OpenAI research collection · 1 manuscript · Source checked 2026-10-19
Colour every point on a flat surface so that two points exactly one unit apart never share a colour. This paper claims five colours cannot work, leaving six or seven as the possible minimum.
The question is easy to state and hard to settle. A proof would narrow a classic problem without yet identifying the exact answer.
Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane.
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
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