Ordinal Definability and Combinatorics of Equivalence Relations

Abstract

Assume $\mathsf{ZF}+\mathsf{A}{\mathsf{D}}^{+}+\mathsf{V}=\mathsf{L}(\mathcal{P}(\mathbb{R}))$. 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. $\mathsf{ZF}+\mathsf{A}{\mathsf{D}}^{+}+\mathsf{V}=\mathsf{L}(\mathcal{P}(\mathbb{R}))$ 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 $f:\mathbb{R}\to \mathbb{R}$ such that for all $x\in \mathbb{R}$, $R(x,f(x))$. $\mathsf{ZF}+\mathsf{AD}$ proves that $\mathbb{R}\times \kappa$ is Jónsson whenever $\kappa$ is an ordinal: For every function $f:[\mathbb{R}\times \kappa {]}_{=}^{<\omega }\to \mathbb{R}\times \kappa$, there is an $A\subseteq \mathbb{R}\times \kappa$ with $A$ in bijection with $\mathbb{R}\times \kappa$ and $f[[A{]}_{=}^{<\omega }]\ne \mathbb{R}\times \kappa$.