Perfect Set Theorems for Equivalence Relations with $I$ - small classes

Abstract

A classical theorem due to Mycielski states that an equivalence relation $E$ having the Baire property and meager equivalence classes must have a perfect set of pairwise inequivalent elements. We consider equivalence relations with $I$-small equivalence classes, where $I$ is a proper $\sigma$-ideal, and ask whether they have a perfect set of pairwise inequivalent elements. We give a positive answer for $E$ universally Baire. We show that the answer for $E$ $\mathbf{\Delta_{2}^{1}}$ is independent of $ZFC$, and find set theoretic assumptions equivalent to it when $I$ is the countable ideal. For equivalence relations which are $\mathbf{\Sigma^1_2}$ and with meager classes, we show that a perfect set of pairwise inequivalent elements exists whenever a Cohen real over $L[z]$ exists for any real $z$ -- which strengthens Mycielski's theorem. A few comments are made about $\sigma$-ideals generated by $\Pi_{1}^{1}$ and orbit equivalence relations.