Invariant uniformization

TL;DR

This paper explores invariant uniformization in descriptive set theory, establishing conditions under which Borel sets admit invariant uniformizations relative to Borel equivalence relations, and classifies the complexity of counterexamples.

Contribution

It characterizes when invariant Borel uniformizations exist for Borel sets under Borel equivalence relations and introduces new proofs and dichotomies extending prior classifications.

Findings

Invariant uniformization exists iff the relation is smooth.

Counterexamples are complex, with $oldsymbol{ riangle}$-completeness for large sections.

New dichotomies generalize Miller and Lecomte's results.

Abstract

Standard results in descriptive set theory provide sufficient conditions for a Borel set $P \subseteq \mathbb{N}^\mathbb{N} \times \mathbb{N}^\mathbb{N}$ to admit a Borel uniformization, namely, when $P$ has "small" sections or "large" sections. We consider an invariant analogue of these results: Given a Borel equivalence relation $E$ and an $E$-invariant Borel set $P$ with "small" or "large" sections, does $P$ admit an $E$-invariant Borel uniformization? For a given Borel equivalence relation $E$, we show that every $E$-invariant Borel set $P$ with "small" or "large" sections admits an $E$-invariant Borel uniformization if and only if $E$ is smooth. We also compute the definable complexity of counterexamples in the case where $E$ is not smooth, using category, measure, and Ramsey-theoretic methods. We provide two new proofs of a dichotomy of Miller classifying the pairs $(E, P)$ such that $P$ admits an $E$-invariant uniformization, for a Borel equivalence relation $E$ and a Borel $E$-invariant set $P$ with countable sections. In the process, we prove an $\aleph_0$-dimensional $(\mathbb{G}_0, \mathbb{H}_0)$ dichotomy, generalizing dichotomies of Miller and Lecomte. We also show that the set of pairs $(E, P)$ such that $P$ has "large" sections and admits an $E$-invariant Borel uniformization is $\boldsymbol{\Sigma^1_2}$-complete; in particular, there is no analog of Miller's dichotomy for $P$ with "large" sections. Finally, we consider a less strict notion of invariant uniformization, where we select a countable nonempty subset of each section instead of a single point.