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Measurable domatic partitions

Edward Hou

TL;DR

This work analyzes when Schreier graphs $\mathrm{Sch}(\Gamma,S,\Gamma)$ arising from finite-dimensional compact Polish groups admit measurable domatic partitions of countably many colors. It develops a framework combining open-cells packing, Gleason–Yamabe dimension theory, and probabilistic tools (Lovász Local Lemma) to construct domatic partitions with open, Borel, or Haar-measurable parts and to lift finite partitions to $\aleph_0$-partitions when $\overline{S}$ is uncountable. Key results include the equivalence of the existence of continuous (and Baire) $\aleph_0$-domatic partitions with the uncountability of $\overline{S}$ for finite-dimensional $\Gamma$, and the universal existence of Haar-measurable domatic partitions for all $S$, along with sum-set applications. The paper further extends these ideas to broader descriptive graph settings, providing both positive constructions and negative results that illuminate the limits and scope of domatic partitions in Borel, Baire, and measure-theoretic contexts.

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.

Measurable domatic partitions

TL;DR

This work analyzes when Schreier graphs arising from finite-dimensional compact Polish groups admit measurable domatic partitions of countably many colors. It develops a framework combining open-cells packing, Gleason–Yamabe dimension theory, and probabilistic tools (Lovász Local Lemma) to construct domatic partitions with open, Borel, or Haar-measurable parts and to lift finite partitions to -partitions when is uncountable. Key results include the equivalence of the existence of continuous (and Baire) -domatic partitions with the uncountability of for finite-dimensional , and the universal existence of Haar-measurable domatic partitions for all , along with sum-set applications. The paper further extends these ideas to broader descriptive graph settings, providing both positive constructions and negative results that illuminate the limits and scope of domatic partitions in Borel, Baire, and measure-theoretic contexts.

Abstract

Let be a compact Polish group of finite topological dimension. For a countably infinite subset , a domatic -partition (for its Schreier graph on ) is a partial function such that for every , one has . We show that a continuous domatic -partition exists, if and only if a Baire measurable domatic -partition exists, if and only if the topological closure of is uncountable. A Haar measurable domatic -partition exists for all choices of . We also investigate domatic partitions in the general descriptive graph combinatorial setting.
Paper Structure (19 sections, 46 theorems, 6 equations)

This paper contains 19 sections, 46 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Proof of Theorem \ref{['thm:1.1']}
  4. Countable compactness implies anti-domaticity
  5. The dimension of a locally compact Polish group
  6. The dimension of a locally compact Polish group
  7. An open-cells packing
  8. An open cover of locally bounded growth
  9. The open pair property and domatic partitions
  10. Application to sum sets
  11. More examples of domatic infinite-partitions
  12. A greedy algorithm on smooth Borel graphs
  13. Measure-theoretic domatic infinite-partitions
  14. Domatic edge-infinite-partitions
  15. More examples of domatic finite partitions
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a finite-dimensional compact Polish group, and let $S\subseteq\Gamma$ be a subset. Then the graph $\mathop{\mathrm{Sch}}\nolimits(\Gamma,S,\Gamma)$ admits a domatic $\aleph_0$-partition with open parts, if and only if it admits a domatic $\aleph_0$-partition with Baire measurable par

Theorems & Definitions (86)

  • Theorem 1.1: Corollary \ref{['cor:2.18']}
  • Theorem 1.2: Corollary \ref{['cor:2.19']}
  • Theorem 1.3: Corollary \ref{['cor:3.6']}
  • Lemma 1.4: Theorem \ref{['thm:2.12']}
  • Theorem 1.5: Corollary \ref{['cor:2.29']}
  • Theorem 1.6: Theorem \ref{['thm:3.5']}
  • Theorem 1.7: Theorem \ref{['thm:4.3']}
  • Theorem 1.8: Theorem \ref{['thm:3.8']}
  • Theorem 1.9: Weilacher, Theorem \ref{['thm:4.5']}
  • Theorem 2.1
  • ...and 76 more