We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in our construction is far milder than earlier ones, and holds in all but the most exotic models of real set theory. On the other hand, we establish productive properties for versions of Menger's property parameterized by filters and semifilters. In particular, the Continuum Hypothesis implies that every productively Menger set of real numbers is productively Hurewicz, and each ultrafilter version of Menger's property is strictly between Menger's and Hurewicz's classic properties. We include a number of open problems emerging from this study.

We provide conceptual proofs of the two most fundamental theorems concerning topological games and open covers: Hurewicz's Theorem concerning the Menger game, and Pawlikowski's Theorem concerning the Rothberger game.