On Sierpi\'nski sets, Hurewicz spaces and Hilgers functions

The Hurewicz property is a classical generalization of $\sigma$-compactness and Sierpi\'nski sets (whose existence follows from CH) are standard examples of non-$\sigma$-compact Hurewicz spaces. We show, solving a problem stated by Szewczak and Tsaban, that for each Sierpi\'nski set S of cardinality at least $\mathfrak b$ there is a Hurewicz space H with $S\times H$ not Hurewicz. Some other questions in the literature concerning this topic are also answered.