Every CBER is smooth below the Carlson–Simpson generic partition
Tom 262 / 2023
Fundamenta Mathematicae 262 (2023), 85-103
MSC: Primary 03E15.
DOI: 10.4064/fm255-12-2022
Opublikowany online: 28 February 2023
Streszczenie
Let 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.
