Perfect independent sets with respect to infinitely many relations

TL;DR

This paper proves the existence of perfect independent sets with respect to infinitely many relations in metric spaces and applies this to show certain free subgroups in Polish groups, answering a recent open question.

Contribution

It introduces a new result on perfect independent sets for countably many relations and applies it to the structure of free subgroups in Polish groups.

Findings

Existence of perfect independent sets for countably many G-delta relations.

Polish groups contain perfect-generated free subgroups if they have uncountable free subgroups.

Answers a recent open question by G{2}b and Strobin.

Abstract

We prove a result on perfect cliques with respect to countably many G-delta relations on a complete metric space. As an application, we show that a Polish group contains a free subgroup generated by a perfect set as long as it contains any uncountable free subgroup. This answers a recent question of G{\l}\c{a}b and Strobin.