Definable expansions on countable groups and countable Borel equivalence relations

TL;DR

This work develops a unified framework for definable expansions of countable structures, bridging expansions on countable Borel equivalence relations (CBER) and definable, equivariant expansions for countable groups. It establishes universal constructions and equivalences between Borel and equivariant expansion problems, and studies random and generic expansions across the Borel, measure, and category settings. The authors apply the general theory to concrete problems such as bijections, Ramsey-type colorings, linearizations, spanning trees, and matchings, proving both positive and negative results depending on the ambient group, the CBER class, and the combinatorial property. The paper also identifies when expansions enforce smoothness versus enabling non-smooth structures, and ends with a suite of open questions on the precise boundaries of expandability and the role of orbit equivalence and hyperfiniteness in this descriptive-combinatorial landscape.

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.

Figures (3)

• Figure 1: There is some $L {\upharpoonright} C_0$-maximal $x_0$ for which $x_0 L' g(x_0)$
• Figure 2: A copy of $\delta {\bm{A}}_1$ in ${\bm{A}}$. The solid arrows are relations in ${\bm{A}}_1$, and the dashed arrows are relations that are forced to exist in $L$.
• Figure 3: Extending $\bar{L}_n$ to $\bar{L}_{n+1}$.

Theorems & Definitions (36)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof : Proof of \ref{['thm:generic-expansions']}