Ordinal Definability and Combinatorics of Equivalence Relations

Abstract: Assume $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$. Let $E$ be a $\mathbf{\Sigma}^1_1$ equivalence relation coded in $\mathrm{HOD}$. $E$ has an ordinal definable equivalence class without any ordinal definable elements if and only if $\mathrm{HOD} \models E$ is unpinned. $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$ proves $E$-class section uniformization when $E$ is a $\mathbf{\Sigma}^1_1$ equivalence relation on $\mathbb{R}$ which is pinned in every transitive model of $\mathsf{ZFC}$ containing the real which codes $E$: Suppose $R$ is a relation on $\mathbb{R}$ such that each section $R_x = {y : (x,y) \in R}$ is an $E$-class, then there is a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x \in \mathbb{R}$, $R(x,f(x))$. $\mathsf{ZF + AD}$ proves that $\mathbb{R} \times \kappa$ is J\'onsson whenever $\kappa$ is an ordinal: For every function $f : [\mathbb{R} \times \kappa]^{<\omega}_= \rightarrow \mathbb{R} \times \kappa$, there is an $A \subseteq \mathbb{R} \times \kappa$ with $A$ in bijection with $\mathbb{R} \times \kappa$ and $f[[A]^{<\omega}_=] \neq \mathbb{R} \times \kappa$.