Every CBER is smooth below the Carlson–Simpson generic partition

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.