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{...
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| creator | Arman, Andrii Bondarenko, Andriy V Prymak, Andriy Radchenko, Danylo |
| description | 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 \(n=48,49\). |
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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 \(n=48,49\).</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Color ; Euclidean geometry ; Euclidean space ; Upper bounds</subject><ispartof>arXiv.org, 2022-02</ispartof><rights>2022. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). 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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 \(n=48,49\).</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><oa>free_for_read</oa></addata></record> |
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| title | Upper bounds on chromatic number of \(\mathbb{E}^n\) in low dimensions |
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