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. This kind of sets were intensively investigated in products of spaces with combinatorial covering properties as Hurewicz, Scheepers, Menger and Rothberger. We will complete this picture with focusing on properties from the second row of the Scheepers diagram. In particular we show that in the Miller model a product space of two $\mathfrak{d}$-concentrated sets has a strong covering property $\mathsf{S}_1(\Gamma,\Omega)$. We provide also counterexamples in products to demonstrate limitations of used methods.