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

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 $E$ ${\mathbf{\Delta }}_{\mathbf{2}}^{\mathbf{1}}$ is independent of $ZFC$, and find set theoretic assumptions equivalent to it when $I$ is the countable ideal. For equivalence relations which are ${\mathbf{\Sigma }}_{\mathbf{2}}^{\mathbf{1}}$ 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 ${\mathrm{\Pi }}_1^1$ and orbit equivalence relations.