Scales, products and the second row of the Scheepers diagram

Abstract We consider products of sets of reals with a combinatorial structure based on scales parameterized by filters. Sets of this kind have been extensively investigated in products of spaces with combinatorial covering properties such as the Hurewicz, Scheepers, Menger, and Rothberger properties. We will complete this picture by focusing on properties from the second row of the Scheepers diagram. We solve in the affirmative a problem due to Bartoszyński–Tsaban, whether all finite powers of a $$\mathfrak {b}$$ b -scale set satisfy the strong covering property $${\textsf{S}_1(\Gamma ,\Omega )}$$ S 1 ( Γ , Ω ) . We also show that, in the Miller model, the product of finitely many $$\mathfrak {d}$$ d -concentrated sets satisfies $${\textsf{S}_1(\Gamma ,\Omega )}$$ S 1 ( Γ , Ω ) . We further provide product counterexamples demonstrating limitations of the methods used.