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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2411.05931 |
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| _version_ | 1866915012174413824 |
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| author | Fiscus, Sean Myzelev, Eric Zhang, Hongyi |
| author_facet | Fiscus, Sean Myzelev, Eric Zhang, Hongyi |
| contents | There is a famous problem in geometric graph theory to find the chromatic number of the unit distance graph on Euclidean space; it remains unsolved. A theorem of Erdos and De-Bruijn simplifies this problem to finding the maximum chromatic number of a finite unit distance graph. Via a construction built on sequential finite graphs obtained from a generalization of this theorem, we have found a class of geometrically defined hypergraphs of arbitrarily large edge cardinality, whose proper colorings exactly coincide with the proper colorings of the unit distance graph on $\mathbb R^d$. We also provide partial generalizations of this result to arbitrary real normed vector spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05931 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A New Class of Geometrically Defined Hypergraphs Arising from the Hadwiger Nelson Problem Fiscus, Sean Myzelev, Eric Zhang, Hongyi Combinatorics There is a famous problem in geometric graph theory to find the chromatic number of the unit distance graph on Euclidean space; it remains unsolved. A theorem of Erdos and De-Bruijn simplifies this problem to finding the maximum chromatic number of a finite unit distance graph. Via a construction built on sequential finite graphs obtained from a generalization of this theorem, we have found a class of geometrically defined hypergraphs of arbitrarily large edge cardinality, whose proper colorings exactly coincide with the proper colorings of the unit distance graph on $\mathbb R^d$. We also provide partial generalizations of this result to arbitrary real normed vector spaces. |
| title | A New Class of Geometrically Defined Hypergraphs Arising from the Hadwiger Nelson Problem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.05931 |