  Which Isbell--Mr\'owka spaces ($\Psi$-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 $\Psi$-space that is (strongly) star-Menger but not star-Hurewicz is
obtained. The PCF-theory function $\kappa\mapsto\cof([\kappa]^\alephes)$ 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.
