When an Equivalence Relation with All Borel Classes will be Borel Somewhere?

TL;DR

The paper investigates conditions under which equivalence relations with all Borel classes become Borel on some large subset, exploring models with large cardinals and determinacy axioms.

Contribution

It establishes new results connecting large cardinal assumptions and determinacy axioms to the Borelness of equivalence relations with all Borel classes.

Findings

Under large cardinal assumptions, such equivalence relations are Borel on some large set.

In models with determinacy axioms, similar Borelness results hold.

The results bridge descriptive set theory, large cardinals, and determinacy hypotheses.

Abstract

In $\mathsf{ZFC}$, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation $E \in L(\mathbb{R})$ on $\mathbb{R}$ with all $\mathbf{\Delta}_1^1$ classes and every $\sigma$-ideal $I$ on $\mathbb{R}$ so that the associated forcing $\mathbb{P}_I$ of $I^+$ $\mathbf{\Delta}_1^1$ subsets is proper, there exists some $I^+$ $\mathbf{\Delta}_1^1$ set $C$ so that $E \upharpoonright C$ is a $\mathbf{\Delta}_1^1$ equivalence relation. In $\mathsf{ZF} + \mathsf{DC} + \mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R}))$, for every equivalence relation $E$ on $\mathbb{R}$ with all $\mathbf{\Delta}_1^1$ classes and every $\sigma$-ideal $I$ on $\mathbb{R}$ so that the associated forcing $\mathbb{P}_I$ is proper, there is some $I^+$ $\mathbf{\Delta}_1^1$ set $C$ so that $E \upharpoonright C$ is a $\mathbf{\Delta}_1^1$ equivalence relation.