Ordinary and measurable colorings of Euclidean three-space

Mikhail Koshelev; Andrey Kupavskii; Vsevolod Voronov

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arXiv · 2610.18907

Ordinary and measurable colorings of Euclidean three-space

Mikhail Koshelev·Andrey Kupavskii·Vsevolod Voronov

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Abstract

A proper coloring of Euclidean space is a coloring that assigns different colors to points at distance one. Using the planar character-law rigidity theorem of OpenAI, we prove that every finite proper coloring of $\R^3$ can be replaced by a proper Lebesgue-measurable coloring with the same number of colors. As a corollary we obtain that the ordinary and measurable chromatic numbers of $\R^3$ are equal, and both lie between $7$ and $15$.

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BibTeX RIS

Mikhail Koshelev, Andrey Kupavskii, Vsevolod Voronov. 2026-10-11. Ordinary and measurable colorings of Euclidean three-space. https://arxiv.org/abs/2610.18907

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