  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.
