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An improved upper bound on the chromatic number of the n-dimensional sphere
Abstract
In this short note, we obtain upper bounds on the chromatic numbers of spheres using recent improvements in bounds for Euclidean space. We show that χ(SnR) ≤ (ξR + o(1))n, where $ξ_R = \frac{3\sqrt 3}{2}(1-1/(4R^2)) $ and χ(SnR) denotes the chromatic number of the n-dimensional sphere of a fixed radius R.
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BibTeX
@misc{bikeev2026improved,
title = {{An improved upper bound on the chromatic number of the n-dimensional sphere}},
author = {A. I. Bikeev and V. A. Voronov and A. M. Raigorodskii},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32408},
eprint = {2610.32408},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32408},
}
RIS
TY - UNPB
TI - An improved upper bound on the chromatic number of the n-dimensional sphere
AU - A. I. Bikeev
AU - V. A. Voronov
AU - A. M. Raigorodskii
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32408
UR - https://arxiv.org/abs/2610.32408
AB - In this short note, we obtain upper bounds on the chromatic numbers of spheres using recent improvements in bounds for Euclidean space. We show that χ(S^n_R) ≤ (ξ_R + o(1))^n, where $ξ_R = \frac{3\sqrt 3}{2}(1-1/(4R^2)) $ and χ(S^n_R) denotes the chromatic number of the n-dimensional sphere of a fixed radius R.
ER -