Extending the Continuum of Six-Colorings
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866910402547286016 |
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| author | Mundinger, Konrad Pokutta, Sebastian Spiegel, Christoph Zimmer, Max |
| author_facet | Mundinger, Konrad Pokutta, Sebastian Spiegel, Christoph Zimmer, Max |
| contents | We present two novel six-colorings of the Euclidean plane that avoid monochromatic pairs of points at unit distance in five colors and monochromatic pairs at another specified distance $d$ in the sixth color. Such colorings have previously been known to exist for $0.41 < \sqrt{2} - 1 \le d \le 1 / \sqrt{5} < 0.45$. Our results significantly expand that range to $0.354 \le d \le 0.657$, the first improvement in 30 years. Notably, the constructions underlying this were derived by formalizing colorings suggested by a custom machine learning approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05509 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extending the Continuum of Six-Colorings Mundinger, Konrad Pokutta, Sebastian Spiegel, Christoph Zimmer, Max Combinatorics We present two novel six-colorings of the Euclidean plane that avoid monochromatic pairs of points at unit distance in five colors and monochromatic pairs at another specified distance $d$ in the sixth color. Such colorings have previously been known to exist for $0.41 < \sqrt{2} - 1 \le d \le 1 / \sqrt{5} < 0.45$. Our results significantly expand that range to $0.354 \le d \le 0.657$, the first improvement in 30 years. Notably, the constructions underlying this were derived by formalizing colorings suggested by a custom machine learning approach. |
| title | Extending the Continuum of Six-Colorings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.05509 |