Borel Polychromatic Number of Grids

TL;DR

This paper investigates Borel polychromatic colorings of grid graphs from free $bZ^d$-actions, establishing the maximum number of colors for such colorings and demonstrating the limits imposed by ergodic actions.

Contribution

It proves that every free $bZ^d$-action admits a Borel $(2^d-1)$-polychromatic coloring, which is optimal under ergodic conditions.

Findings

Every free $bZ^d$-action admits a Borel $(2^d-1)$-polychromatic coloring.

The maximum number of colors in such colorings is $2^d - 1$ for ergodic actions.

The result is sharp; no Borel $2^d$-polychromatic coloring exists under ergodic actions.

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.