Hyperfiniteness on Topological Ramsey Spaces

TL;DR

The paper addresses hyperfiniteness of countable Borel equivalence relations on topological Ramsey spaces, extending the Ellentuck result from $[\,\mathbb{N}\,]^{\mathbb{N}}$ to general Ramsey spaces. It introduces a simple, elementary approach based on sufficiently separated covers and fusion along bounded-degree Borel graphs to canonize colorings and obtain hyperfiniteness on a tail block $[A]$; the method also yields a comeager (generic) version avoiding the Kuratowski–Ulam theorem. The central contribution is a unifying argument that transfers hyperfiniteness from the classic Ellentuck setting to arbitrary topological Ramsey spaces, with a robust toolkit of fusion, KST colorings, and graph separations. The paper concludes with open problems on Ramsey co-null sets and smoothness on positive sets, pointing toward a deeper structural understanding of CBERs in Ramsey-theoretic contexts.

Abstract

We investigate the behavior of countable Borel equivalence relations (CBERs) on topological Ramsey spaces. First, we give a simple proof of the fact that every CBER on $[\mathbb{N}]^{\mathbb{N}}$ is hyperfinite on some set of the form $[A]^{\mathbb{N}}$. Using the idea behind the proof, we show the analogous result for every topological Ramsey space.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Definition 2.1
• Proposition 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1