Definable expansions on countable groups and countable Borel equivalence relations

TL;DR

This paper investigates expansion problems on countable structures within descriptive combinatorics, establishing connections between Borel equivalence relations and countable groups, and providing structure theorems and examples.

Contribution

It introduces a unified framework for expansions on countable structures in Borel, measure, and category settings, linking equivalence relations and groups with new structure theorems.

Findings

Established correspondences between Borel equivalence relations and countable groups.

Proved structure theorems for measure and category settings.

Analyzed examples including spanning trees, Ramsey's theorem, and linearizations.

Abstract

We define and study expansion problems on countable structures in the setting of descriptive combinatorics. We consider both expansions on countable Borel equivalence relations and on countable groups, in the Borel, measure and category settings, and establish some basic correspondences between the two notions. We also prove some general structure theorems for measure and category. We then explore in detail many examples, including finding spanning trees in graphs, finding monochromatic sets in Ramsey's Theorem, and linearizing partial orders.