Scales and combinatorial covering properties

A b-scale set is a subset of P(w) of the form {x_alpha : alpha < b} U Fin, where {x_alpha : alpha < b} is an unbounded set in [w]^w and for all alpha < beta < b we have x_alpha <* x_beta. These sets play a crucial role in the investigation of combinatorial covering properties. Bartoszyński and Shelah showed that each b-scale set is Hurewicz but not sigma-compact which is a counterexample in ZFC for Hurewicz’s conjecture. Under additional set-theoretical assumptions, by the results of Bartoszyński, Tsaban and Weiss all finite powers of a b-scale set are Rothberger and Hurewicz.

Recently, b-scale sets and their generalizations using filters were intensively investigated in products with spaces having Hurewicz, Scheepers or Menger covering properties. Thus far, another classical properties from the second row of the Scheepers Diagram have not been considered in this context. We present new results in this field, in particular we show that in the Miller model a product space of two d-concentrated sets has a strong covering property S_1(Gamma, Omega). We also provide counterexamples in products to demonstrate limitations of used methods.

This is joint work with Piotr Szewczak (Cardinal Stefan Wyszyński University in Warsaw) and Lyubomyr Zdomskyy (Technical University of Vienna).

This research has been completed while the speaker was the Doctoral Candidate in the Interdisciplinary Doctoral School at the Łódź University of Technology, Poland.

The research was supported by the National Science Center, Poland under Weave-UNISONO grant "Set-theoretical aspects of topological selections" 2021/03/Y/ST1/00122.