Equivalence relations which reduce all Borel equivalence relations

We study equivalence relations E such that every Borel equivalence relation is Borel reducible to E, and consider the possibility of there being minimal such relations. We first study the relation of equality of Borel sets, and show that this is not a minimal such relation among co-analytic equivalence relations, nor is it universal for co-analytic relations. We then show that there is no minimum analytic equivalence relation to which all Borel equivalence relations are reducible by producing a “minimal pair ” of analytic equivalence relations with the property that another equivalence relation is reducible to both of these precisely when it is Borel. The theory of definable equivalence relations studies the complexity of equivalence relations defined on Polish spaces under the relation of Borel reducibility. This reducibility notion formalizes the idea of having a definable injection between the corresponding quotient spaces. By a definable equivalenc