  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.
