Measurable domatic partitions

Abstract

Let $Γ$ be a compact Polish group of finite topological dimension. For a countably infinite subset $S\subseteq Γ$, a domatic $\aleph_0$-partition (for its Schreier graph on $Γ$) is a partial function $f:Γ\rightharpoonup\mathbb{N}$ such that for every $x\in Γ$, 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.