Every CBER is smooth below the Carlson-Simpson generic partition

Abstract

Let $E$ be a countable Borel equivalence relation on the space ${\mathcal{E}}_{\infty }$ of all infinite partitions of the natural numbers. We show that $E$ coincides with equality below a Carlson-Simpson generic element of ${\mathcal{E}}_{\infty }$. In contrast, we show that there is a hypersmooth equivalence relation on ${\mathcal{E}}_{\infty }$ which is Borel bireducible with $E_1$ on every Carlson-Simpson cube. Our arguments are classical and require no background in forcing.