Upper Bounds on Chromatic Number of $\mathbb{E}^n$ in Low Dimensions

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 $\mathbb{E}^n$ b...

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Bibliographic Details
Published in:The Electronic journal of combinatorics 2024-05, Vol.31 (2)
Main Authors: Arman, Andrii, Bondarenko, Andriy, Prymak, Andriy, Radchenko, Danylo
Format: Article
Language:English
Subjects:
Combinatorics
Mathematics
Metric Geometry
Online Access:Get full text
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Summary: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 $n\in\{6,8,24\}$, $\chi(\mathbb{E}^7)\le 1372$, $\chi(\mathbb{E}^{9})\leq 17253$, and $\chi(\mathbb{E}^n)\le 3^n$ for all $n\le 38$ and $n\in\{48,49\}$. 
ISSN:1077-8926
1077-8926
DOI:10.37236/11794