On weak Sierpi\'nski sets in groups and free subgroups

TL;DR

This paper proves that groups containing weak Sierpiński sets necessarily contain free subgroups, confirming a conjecture and improving previous results with simplified proofs.

Contribution

It establishes that the existence of weak Sierpiński sets in groups implies the presence of free subgroups, confirming a conjecture and refining prior proofs.

Findings

Groups with weak Sierpiński sets contain free subgroups

The conjecture by Tomkowicz and Wagon is proven

Proofs are simplified and shortened

Abstract

In this paper we discuss the problem of existence of so called weak Sierpi\'nski sets in groups. It is known that group $G$ has a Sierpi\'nski subset if and only if it contains a free subgroup. In their paper, Tomkowicz and Wagon conjectured that an analogous result holds also for the weaker condition. We derive a number of properties of groups with weak Sierpi\'nski subsets and use them to prove the above mentioned conjecture. This is an improved version of arXiv:1805.11486. Some of the included proofs have been simplified and shortened.