Upper bounds on chromatic number of $\mathbb{E}^n$ in low dimensions
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| Main Authors: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866910909800120320 |
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| author | Arman, Andrii Bondarenko, Andriy V. Prymak, Andriy Radchenko, Danylo |
| author_facet | Arman, Andrii Bondarenko, Andriy V. Prymak, Andriy Radchenko, Danylo |
| contents | 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 $n=48,49$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_13438 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Upper bounds on chromatic number of $\mathbb{E}^n$ in low dimensions Arman, Andrii Bondarenko, Andriy V. Prymak, Andriy Radchenko, Danylo Combinatorics Metric Geometry 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 $n=48,49$. |
| title | Upper bounds on chromatic number of $\mathbb{E}^n$ in low dimensions |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2112.13438 |