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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| Veröffentlicht in: | The Electronic journal of combinatorics 2024-05, Vol.31 (2) |
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| container_title | The Electronic journal of combinatorics |
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| creator | Arman, Andrii Bondarenko, Andriy 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 $n\in\{48,49\}$. |
| doi_str_mv | 10.37236/11794 |
| format | Article |
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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 $n\in\{48,49\}$. </description><identifier>ISSN: 1077-8926</identifier><identifier>EISSN: 1077-8926</identifier><identifier>DOI: 10.37236/11794</identifier><language>eng</language><publisher>Open Journal Systems</publisher><subject>Combinatorics ; Mathematics ; Metric Geometry</subject><ispartof>The Electronic journal of combinatorics, 2024-05, Vol.31 (2)</ispartof><rights>Attribution</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><orcidid>0000-0003-1457-6214</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>230,314,776,780,860,881,27901,27902</link.rule.ids><backlink>$$Uhttps://hal.science/hal-04789024$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Arman, Andrii</creatorcontrib><creatorcontrib>Bondarenko, Andriy</creatorcontrib><creatorcontrib>Prymak, Andriy</creatorcontrib><creatorcontrib>Radchenko, Danylo</creatorcontrib><title>Upper Bounds on Chromatic Number of $\mathbb{E}^n$ in Low Dimensions</title><title>The Electronic journal of combinatorics</title><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. 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| source | DOAJ Directory of Open Access Journals; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | Combinatorics Mathematics Metric Geometry |
| title | Upper Bounds on Chromatic Number of $\mathbb{E}^n$ in Low Dimensions |
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