Scales and combinatorial covering properties

Abstract: A b-scale set is a subset of P(ω) of the form {xα : α < b}∪Fin, where {xα : α < b} is an
unbounded set in [ω]^ω and for all α < β < b we have x_α ≤∗ x_β. These sets play a crucial
role in the investigation of combinatorial covering properties. Bartoszynski and Shelah
showed that each b-scale set is Hurewicz but not σ-compact which is a counterexample
in ZFC for Hurewicz’s conjecture. Under additional set-theoretical assumptions, by the
results of Bartoszy ́nski, Tsaban, Fremlin, Miller and Scheepers 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 S1(Γ, Ω). We also provide counterexamples in products to demonstrate
limitations of used methods.