Invariant uniformization

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.