A set of reals $X$ is $\mathfrak{b}$-concentrated if it has cardinality at least $\mathfrak{b}$ and it contains a countable set $D\subseteq X$ such that each closed subset of $X$ disjoint from $D$ has size smaller than $\mathfrak{b}$. $\kappa$-concentrated sets play a crucial role in the investigation of combinatorial covering properties. It is independent of ZFC if all $\mathfrak{b}$-concentrated sets are Hurewicz. We present ZFC results about how structures of $\mathfrak{b}$-concentrated sets with the Hurewicz covering property can be characterized with the notion of being meager-unbounded, which we then use to get that such structures are productively Hurewicz. We obtain that assuming that the semifilter trichotomy holds each $\mathfrak{b}$-concentrated set is Hurewicz and even productively Hurewicz. In particular, in the Miller model $\mathfrak{b}$-concentrated sets are also productively Rothberger. We analyze the Laver model, where the behavior of Hurewicz $\mathfrak{b}$-concentrated sets differs from the one under the semifilter trichotomy.

This is joint work with Piotr Szewczak (University of Warsaw) and Lyubomyr Zdomskyy (TU Vienna).