Measurability and Perfect Set Theorems for Equivalence Relations with Small Classes

Abstract

We ask whether $\mathbf{\Delta^1_2}$ or $\mathbf{\Sigma^1_2}$ equivalence relations with $I$-small classes for $I$ a $\sigma$-ideal must have perfectly many classes. We show that for a wide class of ccc $\sigma$-ideals, a positive answer for $\mathbf{\Delta^1_2}$ equivalence relations is equivalent to the $I$-measurability of $\mathbf{\Delta^1_2}$ sets. However, the analogous statement for $\mathbf{\Sigma^1_2}$ equivalence relations is false: $\mathbf{\Sigma^1_2}$ equivalence relations with meager classes have a perfect set of pairwise inequivalent elements if and only if $\mathbf{\Delta^1_2}$ sets have the Baire property.