Every CBER is smooth below the CarlsonSimpson 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 CarlsonSimpson 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 CarlsonSimpson cube. Our arguments are classical and require no background in forcing.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 arXiv:
 arXiv:2206.14224
 Bibcode:
 2022arXiv220614224P
 Keywords:

 Mathematics  Logic;
 Mathematics  Combinatorics;
 Mathematics  Dynamical Systems;
 03E15