Upper bounds on chromatic number of $\mathbb{E}^n$ in low dimensions
Andrii Arman, Andriy V. Bondarenko, Andriy Prymak, Danylo Radchenko
TL;DR
The paper advances the study of χ(E^n) in low dimensions by developing and applying sublattice coloring schemes tied to lattice Voronoi cells and covering-packing ratios, augmented with computer-assisted constructions. It establishes new explicit upper bounds: χ(E^5) ≤ 140, χ(E^n) ≤ 7^{n/2} for n ∈ {2,4,6,8,24}, χ(E^7) ≤ 1372, χ(E^9) ≤ 17253, and χ(E^n) ≤ 3^n for all n ≤ 38 and for n = 48,49; these bounds are derived from both linear/sub-lattice coloring arguments and Eisenstein or laminated-lattice frameworks. The work also details computational verification methods, explores strategies to search sublattices efficiently, and discusses lower bounds within small dimensions, highlighting the limits and potential of these coloring approaches. Overall, the results provide a suite of tight, dimension-specific upper bounds and demonstrate the viability of lattice-based, computer-assisted methods for refining χ(E^n) in higher dimensions.
Abstract
Let $χ(\mathbb{E}^n)$ 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: $χ(\mathbb{E}^5)\le 140$, $χ(\mathbb{E}^n)\le 7^{n/2}$ for $n\in\{6,8,24\}$, $χ(\mathbb{E}^7)\le 1372$, $χ(\mathbb{E}^{9})\leq 17253$, and $χ(\mathbb{E}^n)\le 3^n$ for all $n\le 38$ and $n=48,49$.