The Euclidean plane is not five-colorable
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.
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In one of OpenAI's recent papers , there is a claim that the unit distance graph of the euclidean plane isn't 5-colorable , which is major progress on the Hadwiger–Nelson problem ( assuming the paper isn't retracted ). This result implies, by the De Bruijn–Erdős theorem or the contrapositive of…
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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
raises the lower bound to six, leaving six and seven as the two possible answers. The Kakeya problem. My post on Hong Wang and Joshua Zahl’s breakthrough discussed the three-dimensional Kakeya set conjecture and the stronger maximal-function problem. The release announces the maximal conjecture in…