Strong ergodicity around countable products of countable equivalence relations

Abstract: This paper deals with countable products of countable Borel equivalence relations and equivalence relations "just above" those in the Borel reducibility hierarchy. We show that if $E$ is strongly ergodic with respect to $\mu$ then $E^\mathbb{N}$ is strongly ergodic with respect to $\mu^\mathbb{N}$. We answer questions of Clemens and Coskey regarding their recently defined $\Gamma$-jump operations, in particular showing that the $\mathbb{Z}^2$-jump of $E_\infty$ is strictly above the $\mathbb{Z}$-jump of $E_\infty$. We study a notion of equivalence relations which can be classified by infinite sequences of "definably countable sets". In particular, we define an interesting example of such equivalence relation which is strictly above $E_\infty^\mathbb{N}$, strictly below $=^+$, and is incomparable with the $\Gamma$-jumps of countable equivalence relations. We establish a characterization of strong ergodicity between Borel equivalence relations in terms of symmetric models. The proofs then rely on a fine analysis of the very weak choice principles "every sequence of $E$-classes admits a choice sequence", for various countable Borel equivalence relations $E$.