Hyperfiniteness on Topological Ramsey Spaces

TL;DR

This paper proves that all countable Borel equivalence relations on topological Ramsey spaces are hyperfinite on certain subsets, extending known results from classical spaces to a broader class of topological Ramsey spaces.

Contribution

It provides a unified proof that CBERs are hyperfinite on specific subsets across all topological Ramsey spaces, generalizing previous results.

Findings

CBERs on $[ abla]^{ abla}$ are hyperfinite on some $[A]^{ abla}$

The result extends to all topological Ramsey spaces

Introduces a simple proof technique for hyperfiniteness

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.