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arXiv · 2508.18559

Borel Polychromatic Number of Grids

Katalin Berlow·Edward Hou
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Abstract

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.

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Katalin Berlow, Edward Hou. 2025-08-25. Borel Polychromatic Number of Grids. https://arxiv.org/abs/2508.18559

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