Dichotomy Theorems for Families of Non-Cofinal Essential Complexity

TL;DR

This paper establishes a dichotomy for Borel equivalence relations, showing that each is either reducible to a simple relation or has a complex family of incompatible relations, with implications for the structure of Borel reducibility.

Contribution

It proves a new dichotomy theorem for Borel equivalence relations, characterizing the complexity and incompatibility within the Borel reducibility hierarchy.

Findings

Either $E$ is reducible to $ ext{E}_0$ or incompatible relations have cofinal essential complexity.

Families of non-cofinal complexity are only composed of smooth relations under certain conditions.

The dichotomy aligns with and extends known theorems in Borel reducibility.

Abstract

We prove that for every Borel equivalence relation $E$, either $E$ is Borel reducible to $\mathbb{E}\_0$, or the family of Borel equivalence relations incompatible with $E$ has cofinal essential complexity. It follows that if $F$ is a Borel equivalence relation and $\cal F$ is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation $E$, either $E\in {\cal F}$ or $F$ is Borel reducible to $E$, then $\cal F$ consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.