$F_\sigma$ equivalence relations and Laver forcing

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.