A new proof of the Larman-Rogers upper bound for the chromatic number of the Euclidean space

Abstract

The chromatic number $\chi ({\mathbb{R}}^n)$ of the Euclidean space ${\mathbb{R}}^n$ is the smallest number of colors sufficient for coloring all points of the space in such a way that any two points at the distance 1 have different colors. In 1972 Larman--Rogers proved that $\chi ({\mathbb{R}}^n)\le (3+o(1){)}^n$. We give a new proof of this bound.