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

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 $n\in \{6,8,24\}$, $\chi ({\mathbb{E}}^7)\le 1372$, $\chi ({\mathbb{E}}^9)\le 17253$, and $\chi ({\mathbb{E}}^n)\le 3^n$ for all $n\le 38$ and $n=48,49$.

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