$F_\sigma$ equivalence relations and Laver forcing

TL;DR

This paper investigates Borel equivalence relations on Laver trees, demonstrating that those reducible to certain $F_\sigma$ $P$-ideal-based relations can be simplified to either the full or identity relation, with implications for dichotomies in descriptive set theory.

Contribution

It establishes a canonization result for Borel equivalence relations on Laver trees related to $F_\sigma$ $P$-ideals, extending the understanding of their structure.

Findings

Equivalence relations reducible to $F_\sigma$ $P$-ideal relations can be canonized to simple forms.

Proves Silver type dichotomy for the Laver ideal.

Provides a classification framework for Borel equivalences on Laver trees.

Abstract

Following the topic of the book Canonical Ramsey Theory on Polish Spaces by V. Kanovei, M. Sabok and J. Zapletal we study Borel equivalences on Laver trees. Here we prove that equivalence relations Borel reducible to an equivalence relation on $2^\omega$ given by some $F_\sigma$ $P$-ideal on $\omega$ can be canonized to the full equivalence relation or to the identity relation. This has several corollaries, e.g. Silver type dichotomy for the Laver ideal and equivalences Borel reducible to equivalence relations given by $F_\sigma$ $P$-ideals.