Dichotomy Theorems for Families of Non-Cofinal Essential Complexity
We prove that for every Borel equivalence relation E, either E is Borel reducible to 𝔼0, 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 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 ∈ ℱ or F is Borel reducible to E, then ℱ consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.
Clemens, John D.; Lecomte, Dominique; and Miller, Benjamin D. (2017). "Dichotomy Theorems for Families of Non-Cofinal Essential Complexity". Advances in Mathematics, 304, 285-299. http://dx.doi.org/10.1016/j.aim.2016.08.044