Borel Polychromatic Number of Grids
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908503763845120 |
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| author | Berlow, Katalin Hou, Edward |
| author_facet | Berlow, Katalin Hou, Edward |
| contents | We study Borel polychromatic colorings of grid graphs arising from free Borel actions of $\mathbb{Z}^d$. A polychromatic coloring is one in which every unit $d$-dimensional cube sees all available colors. In the classical setting, every grid admits a $2^d$-polychromatic coloring, while in the Borel setting this fails. Our main result shows that every free $\mathbb{Z}^d$-action admits a Borel $(2^d-1)$-polychromatic coloring. This result is sharp: any action where the generators act ergodically does not admit a Borel $2^d$-polychromatic coloring. We conclude with open directions for extending the theory beyond cube tilings and for exploring the dependence of Borel polychromatic numbers on the underlying action. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18559 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Borel Polychromatic Number of Grids Berlow, Katalin Hou, Edward Logic Combinatorics 03E15 (Primary), 05C15 (Secondary) We study Borel polychromatic colorings of grid graphs arising from free Borel actions of $\mathbb{Z}^d$. A polychromatic coloring is one in which every unit $d$-dimensional cube sees all available colors. In the classical setting, every grid admits a $2^d$-polychromatic coloring, while in the Borel setting this fails. Our main result shows that every free $\mathbb{Z}^d$-action admits a Borel $(2^d-1)$-polychromatic coloring. This result is sharp: any action where the generators act ergodically does not admit a Borel $2^d$-polychromatic coloring. We conclude with open directions for extending the theory beyond cube tilings and for exploring the dependence of Borel polychromatic numbers on the underlying action. |
| title | Borel Polychromatic Number of Grids |
| topic | Logic Combinatorics 03E15 (Primary), 05C15 (Secondary) |
| url | https://arxiv.org/abs/2508.18559 |