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

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.