  We investigate products of sets of reals with combinatorial covering
properties. A topological space satisfies $\mathsf{S}_1(\Gamma,\Gamma)$ if for
each sequence of point-cofinite open covers of the space, one can pick one
element from each cover and obtain a point-cofinite cover of the space. We
prove that, if there is an unbounded tower, then there is a nontrivial set of
reals satisfying $\mathsf{S}_1(\Gamma,\Gamma)$ in all finite powers. In
contrast to earlier results, our proof does not require any additional
set-theoretic assumptions.
  A topological space satisfies $\Omega\choose\Gamma$ (also known as
Gerlits--Nagy's property $\gamma$) if every open cover of the space such that
each finite subset of the space is contained in a member of the cover, contains
a point-cofinite cover of the space. We investigate products of sets satisfying
$\Omega\choose\Gamma$ and their relations to other classic combinatorial
covering properties. We show that finite products of sets with a certain
combinatorial structure satisfy $\Omega\choose\Gamma$ and give necessary and
sufficient conditions when these sets are productively $\Omega\choose\Gamma$.
