Combinatorial aspects of selective star covering properties in Ψ-spaces

Which Isbell–Mrówka spaces (Ψ-spaces) satisfy the star version of Menger's and Hurewicz's covering properties? Following Bonanzinga and Matveev, this question is considered here from a combinatorial point of view. An example of a Ψ-space that is (strongly) star-Menger but not star-Hurewicz is obtained. The PCF-theory function κ↦cof([κ]ℵ0)κ↦cof([κ]ℵ0) is a key tool. Using the method of forcing, a complete answer to a question of Bonanzinga and Matveev is provided.The results also apply to the mentioned covering properties in the realm of Pixley–Roy spaces, to the extent of spaces with these properties, and to the character of free abelian topological groups over hemicompact k spaces.