Upper bounds on chromatic number of En in low dimensions

Abstract

Let χ(En) denote the chromatic number of the Euclidean space En, i.e., the smallest number of colors that can be used to color En so that no two points unit distance apart are of the same color. We present explicit constructions of colorings of En based on sublattice coloring schemes that establish the following new bounds: χ(E5)≤140, χ(En)≤7n/2 for n∈{6,8,24}, χ(E7)≤1372, χ(E9)≤17253, and χ(En)≤3n for all n≤38 and n∈{48,49}.