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September 10, 2025

The Borel Ramsey properties for countable Borel equivalence relations

SGSu GaoNankai University MXMing XiaoNankai University

Key Points

A countable Borel equivalence relation is smooth if and only if strong and weak Borel Ramsey properties hold.
The findings indicate that smooth equivalence relations exhibit specific structural properties.
Variations of the Borel Ramsey properties are explored for hyperfinite non-smooth Borel equivalence relations.
Understanding these properties could advance theoretical aspects in 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.

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Cite This Study

Gao et al. (Fri,) studied this question.

synapsesocial.com/papers/68c1d7ee54b1d3bfb60fa059 https://doi.org/https://doi.org/10.1090/proc/17340