Upper bounds on chromatic number of $\mathbb{E}^n$ in low dimensions

TL;DR

This paper introduces new explicit bounds on the chromatic number of Euclidean spaces in low dimensions using sublattice coloring schemes, improving previous estimates for specific dimensions.

Contribution

It provides novel explicit constructions of colorings that establish improved upper bounds on the chromatic number of Euclidean spaces in certain low dimensions.

Findings

(\mathbb{E}^5)   140

(\mathbb{E}^n)  7^{n/2} for n=6,8,24

(\mathbb{E}^7)  1372

Abstract

Let $\chi(\mathbb{E}^n)$ denote the chromatic number of the Euclidean space $\mathbb{E}^n$, i.e., the smallest number of colors that can be used to color $\mathbb{E}^n$ so that no two points unit distance apart are of the same color. We present explicit constructions of colorings of $\mathbb{E}^n$ based on sublattice coloring schemes that establish the following new bounds: $\chi(\mathbb{E}^5)\le 140$, $\chi(\mathbb{E}^n)\le 7^{n/2}$ for $n\in\{6,8,24\}$, $\chi(\mathbb{E}^7)\le 1372$, $\chi(\mathbb{E}^{9})\leq 17253$, and $\chi(\mathbb{E}^n)\le 3^n$ for all $n\le 38$ and $n=48,49$.