Structural results on idealistic equivalence relations

TL;DR

This paper explores the structure of idealistic equivalence relations, demonstrating the existence of many non-isomorphic relations under certain conditions and providing counterexamples to longstanding conjectures.

Contribution

It shows that under analytic determinacy, there are continuum many idealistic analytic equivalence relations not classwise Borel isomorphic to orbit relations, and offers a counterexample to a variant of the E1 conjecture.

Findings

Existence of continuum many non-isomorphic idealistic analytic equivalence relations

Counterexample to a variant of the E1 conjecture from 1997

Alternative formulations of the E1 conjecture discussed

Abstract

We address some fundamental problems concerning the structure of idealistic equivalence relations. In particular, we show that, under analytic determinacy, there are continuum many idealistic analytic equivalence relations that are not classwise Borel isomorphic to an orbit equivalence relation. Moreover, we also discuss alternative formulations of the E1 conjecture, and present an elementary counterexample to one of its earliest variants proposed by Hjorth and Kechris in 1997.