Measurable domatic partitions

Abstract

Let $\hat{I}“$ be a compact Polish group of finite topological dimension. For a countably infinite subset $S\subseteq \hat{I}“$, a domatic ${\aleph }_0$-partition (for its Schreier graph on $\hat{I}“$) is a partial function $f:\hat{I}“\rightharpoonup \mathbb{N}$ such that for every $x\in \hat{I}“$, 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.

35 pages. This version supersedes both the previous version and arXiv:2209.14534. To appear in the Journal of Symbolic Logic