Ramsey and Hypersmoothness

TL;DR

This paper presents a new example of an $F_\sigma$ equivalence relation that is not hypersmooth, expanding the understanding of the structure of such relations in descriptive set theory.

Contribution

It combines existing canonization results to construct a natural example of an $F_\sigma$ relation lacking hypersmoothness, which was previously unknown.

Findings

Provides the first natural example of a non-hypersmooth $F_\sigma$ relation.

Integrates results from Pr"omel-Voigt, Mathias, and Soare.

Advances the classification of equivalence relations in descriptive set theory.

Abstract

Combining canonization results of Pr\"omel-Voigt, Mathias, and Soare, we provide a new, natural example of an $F_\sigma$ equivalence relation that is not hypersmooth.