Monochromatic infinite sets in Minkowski planes
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866915515085094912 |
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| author | Frankl, Nóra Gehér, Panna Sagdeev, Arsenii Tóth, Géza |
| author_facet | Frankl, Nóra Gehér, Panna Sagdeev, Arsenii Tóth, Géza |
| contents | We prove that for any $\ell_p$-norm in the plane with $1<p<\infty$ and for every infinite $\mathcal{M} \subset \mathbb{R}^2$, there exists a two-colouring of the plane such that no isometric copy of $\mathcal{M}$ is monochromatic. On the contrary, we show that for every polygonal norm (that is, the unit ball is a polygon) in the plane, there exists an infinite $\mathcal{M} \subset \mathbb{R}^2$ such that for every two-colouring of the plane there exists a monochromatic isometric copy of $\mathcal{M}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_08840 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Monochromatic infinite sets in Minkowski planes Frankl, Nóra Gehér, Panna Sagdeev, Arsenii Tóth, Géza Combinatorics Metric Geometry 05C15, 05D10, 52C10 We prove that for any $\ell_p$-norm in the plane with $1<p<\infty$ and for every infinite $\mathcal{M} \subset \mathbb{R}^2$, there exists a two-colouring of the plane such that no isometric copy of $\mathcal{M}$ is monochromatic. On the contrary, we show that for every polygonal norm (that is, the unit ball is a polygon) in the plane, there exists an infinite $\mathcal{M} \subset \mathbb{R}^2$ such that for every two-colouring of the plane there exists a monochromatic isometric copy of $\mathcal{M}$. |
| title | Monochromatic infinite sets in Minkowski planes |
| topic | Combinatorics Metric Geometry 05C15, 05D10, 52C10 |
| url | https://arxiv.org/abs/2308.08840 |