Every CBER is smooth below the Carlson-Simpson generic partition

2022/06/28 by Aristotelis Panagiotopoulos, Panagiotopoulos, Aristotelis, Allison Wang +1
Mathematics · #Advanced Topology and Set Theory #Functional Equations Stability Results #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2206.14224

Abstract

Let E be a countable Borel equivalence relation on the space E_∞ of all infinite partitions of the natural numbers. We show that E coincides with equality below a Carlson-Simpson generic element of E_∞. In contrast, we show that there is a hypersmooth equivalence relation on E_∞ which is Borel bireducible with E₁ on every Carlson-Simpson cube. Our arguments are classical and require no background in forcing.

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