Every CBER is smooth below the Carlson-Simpson generic partition

TL;DR

The paper demonstrates that any countable Borel equivalence relation on the space of infinite partitions becomes trivial below a Carlson-Simpson generic element, contrasting with the existence of a hypersmooth relation that remains complex on all Carlson-Simpson cubes.

Contribution

It proves that all countable Borel equivalence relations are smooth below Carlson-Simpson generic partitions, and constructs a hypersmooth relation with persistent complexity on these cubes.

Findings

Countable Borel relations are smooth below Carlson-Simpson generic elements.

Existence of a hypersmooth relation Borel bireducible with $E_1$ on every Carlson-Simpson cube.

Classical arguments used without forcing techniques.

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.