Geodesic
Upper bounds on chromatic number of \(\mathbb{E}^n\) in low dimensions
The electronic journal of combinatorics, Tome 31 (2024) no. 2
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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 $n\in\{48,49\}$.
DOI : 10.37236/11794
Classification : 05C15, 11H31, 05B40, 52C17
Mots-clés : sublattice coloring schemes, computer assistance 
@article{10_37236_11794,
     author = {Andrii Arman and Andriy Bondarenko and Andriy Prymak and Danylo Radchenko},
     title = {Upper bounds on chromatic number of {\(\mathbb{E}^n\)} in low dimensions},
     journal = {The electronic journal of combinatorics},
     year = {2024},
     volume = {31},
     number = {2},
     doi = {10.37236/11794},
     zbl = {1543.05046},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/11794/}
}
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Andrii Arman; Andriy Bondarenko; Andriy Prymak; Danylo Radchenko. Upper bounds on chromatic number of \(\mathbb{E}^n\) in low dimensions. The electronic journal of combinatorics, Tome 31 (2024) no. 2. doi: 10.37236/11794

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