%0 Journal Article
%T Dichotomy Theorems for Families of Non-Cofinal Essential Complexity
%A John D. Clemens
%A Dominique Lecomte
%A Benjamin D. Miller
%J Mathematics
%D 2014
%I arXiv
%X 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.
%U http://arxiv.org/abs/1412.8684v1