Concentrated sets and the Hurewicz property

Abstract

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 with $D$ has size smaller than $\mathfrak{b}$. We present ZFC results about structures of $\mathfrak{b}$-concentrated sets with the Hurewicz covering property using semifilters. Then we show that assuming that the semifilter trichotomy holds, then each $\mathfrak{b}$-concentrated set is Hurewicz and even productively Hurewicz. We also show that the appearance of Hurewicz $\mathfrak{b}$-concentrated sets under the semifilter trichotomy is somewhat specific and the situation in the Laver model for the consitency of the Borel Conjecture is different.