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The Borel Ramsey properties for countable Borel equivalence relations

Su Gao, Ming Xiao

TL;DR

This work characterizes when invariant Borel Ramsey properties hold for countable Borel equivalence relations by tying them to smoothness. It defines strong and weak Borel Ramsey properties using Borel $E$-colorings and complete sections, proving that for aperiodic $E$ these properties hold for some/all $n,k>1$ if and only if $E$ is smooth, with $n=1$ treated as a base case. The paper further develops a hyperfinite nonsmooth extension via almost transitive colorings, showing that hyperfinite $E$ admit Borel monochromatic reductions to $E_1$ for such colorings, while providing a counterexample to reductions in general. The methods combine Feldman–Moore representations, the Glimm–Effros dichotomy, transversals, and explicit constructions to encode Ramsey witnesses as Borel objects, clarifying the boundary between smooth, hyperfinite, and nonsmooth cases in invariant descriptive set theory.

Abstract

We define some natural notions of strong and weak Borel Ramsey properties for countable Borel equivalence relations and show that they hold for a countable Borel equivalence relation if and only if the equivalence relation is smooth. We also consider some variation of the notion for hyperfinite non-smooth Borel equivalence relations.

The Borel Ramsey properties for countable Borel equivalence relations

TL;DR

This work characterizes when invariant Borel Ramsey properties hold for countable Borel equivalence relations by tying them to smoothness. It defines strong and weak Borel Ramsey properties using Borel -colorings and complete sections, proving that for aperiodic these properties hold for some/all if and only if is smooth, with treated as a base case. The paper further develops a hyperfinite nonsmooth extension via almost transitive colorings, showing that hyperfinite admit Borel monochromatic reductions to for such colorings, while providing a counterexample to reductions in general. The methods combine Feldman–Moore representations, the Glimm–Effros dichotomy, transversals, and explicit constructions to encode Ramsey witnesses as Borel objects, clarifying the boundary between smooth, hyperfinite, and nonsmooth cases in invariant descriptive set theory.

Abstract

We define some natural notions of strong and weak Borel Ramsey properties for countable Borel equivalence relations and show that they hold for a countable Borel equivalence relation if and only if the equivalence relation is smooth. We also consider some variation of the notion for hyperfinite non-smooth Borel equivalence relations.
Paper Structure (3 sections, 13 theorems, 37 equations)

This paper contains 3 sections, 13 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Proof of the Main Theorem
  3. Hyperfinite Nonsmooth Equivalence Relations

Key Result

Theorem 1.1

For any integer $n,k>0$ and function $c: [\mathbb{N}]^n\to \{1,\dots, k\}$, there exists $a\in\{1,\dots, k\}$ and an infinite subset $M\subseteq\mathbb{N}$ such that for all $x\in [M]^n$, $c(x)=a$.

Theorems & Definitions (28)

  • Theorem 1.1: Ramsey
  • Definition 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 18 more