arXiv subjects
Publications and source records attributed to Sławomir Szczepaniak.
A combinatorial proof of a pigeonhole principle of Gowers is found along with its symmetric and approximate version, FIN$_k^\pm$ theorem. The proofs do not use of the concept of ultrafilter.
It is proved that no non-meager subspace of the space $[ω]^ω$ equipped with the Ellentuck topology does admit a Kuratowski partition, that is such a subset cannot be covered by a family $\mathfrak{F}$ of disjoint relatively meager sets such that $\bigcup\mathfrak{F}'$ has the Baire property (relatively) for every subfamily $\mathfrak{F}'\subseteq\mathfrak{F}$. It is also shown that the existence of Kuratowski partitions is not a metric problem. Some remarks concerning continuous restrictions of functions with domain in the Ellentuck space are made.
The structure of quotient Boolean algebras in terms of cardinal invariants is investigated. Some results of Gitik and Shelah regarding atomless ideals are reproved and proofs are significantly simplified.