Upper Bounds on Chromatic Number of Eⁿ in Low Dimensions

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\.