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 $f[[A]^{ <\omega}_=] \neq \mathbb{ R} \times \kappa$.
Categories: math.LO