PreprintCC BY 4.0arXiv
Ordinary and measurable colorings of Euclidean three-space
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 R3 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 R3 are equal, and both lie between $7$ and $15$.
Related
- Rigidity of Euclidean Convex Fillings for Ordinary Ambrosio--Kirchheim Mass
- On one-space dimensional parabolic equations with measurable coefficients: Sobolev estimates and the Alexandrov maximum principle
- Free Boundary Minimal surfaces in the Euclidean Three-Ball close to the boundary
- Pseudoconvex domains in complex Euclidean space with finite-dimensional Bergman spaces
- Foliation by bi-rotational self shrinkers in Euclidean space
BibTeX
@misc{koshelev2026ordinary,
title = {{Ordinary and measurable colorings of Euclidean three-space}},
author = {Mikhail Koshelev and Andrey Kupavskii and Vsevolod Voronov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18907},
eprint = {2610.18907},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18907},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ordinary and measurable colorings of Euclidean three-space
AU - Mikhail Koshelev
AU - Andrey Kupavskii
AU - Vsevolod Voronov
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.18907
UR - https://arxiv.org/abs/2610.18907
AB - 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 $15$.
ER -