Dichotomy Theorems for Families of Non-Cofinal Essential Complexity

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 $\mathcal{F}$ 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 $E\in \mathcal{F}$ or $F$ is Borel reducible to $E$, then $\mathcal{F}$ consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.