The Scheepers property and products of Menger spaces

A topological space X is Menger if for every sequence of open covers O_1, O_2,... of the space X, there are finite subfamilies F_1\subseteq O_1, F_2\subseteq O_2,... such that their union is a cover of X. If, in addition, for every finite subset F of X there is a natural number n with F\subseteq \bigcup F_n, then the space X is Scheepers. The above properties generalize \sigma-compactness, and Scheepers' property is formally stronger than Menger's property. It is consistent with ZFC that these properties are equal. One of the open problems in the field of selection principles is to find the minimal hypothesis that the above properties can be separated in the class of sets of reals. Using purely combinatorial approach, we provide examples under some set theoretic hypotheses. We apply obtained results to products of Menger spaces. This a joint work with Boaz Tsaban (Bar-Ilan University, Israel) and Lyubomyr Zdomskyy (Kurt Goedel Research Center, Austria).