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MEASURABLE DOMATIC PARTITIONS
Published online by Cambridge University Press: 07 August 2025
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.
MSC classification
Primary:
03E15: Descriptive set theory
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
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