OpenAlex ·
Ordinary and measurable colorings of Euclidean three-space
- Published
- Source
- OpenAlex
- Published in
- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2610.18907
- Original
- Read it at the publisher
- Topics
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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