Measurable domatic partitions

TL;DR

This paper characterizes when continuous, Baire measurable, and Haar measurable domatic partitions exist in compact Polish groups, linking their existence to the topological closure of a subset and exploring their properties in descriptive graph settings.

Contribution

It establishes the equivalence between the existence of continuous and Baire measurable domatic partitions and the uncountability of the closure of the subset, also providing conditions for Haar measurable partitions.

Findings

Continuous and Baire measurable partitions are equivalent under certain conditions.

Haar measurable partitions exist universally for all subsets.

Topological closure of the subset determines the existence of partitions.

Abstract

Let $\Gamma$ be a compact Polish group of finite topological dimension. For a countably infinite subset $S\subseteq \Gamma$, a domatic $\aleph_0$-partition (for its Schreier graph on $\Gamma$) is a partial function $f:\Gamma\rightharpoonup\mathbb{N}$ such that for every $x\in \Gamma$, one has $f[S\cdot x]=\mathbb{N}$. We show that a continuous domatic $\aleph_0$-partition exists, if and only if a Baire measurable domatic $\aleph_0$-partition exists, if and only if the topological closure of $S$ is uncountable. A Haar measurable domatic $\aleph_0$-partition exists for all choices of $S$. We also investigate domatic partitions in the general descriptive graph combinatorial setting.