Bartoszyński and Halbeisen conjectured that in the Miller model there exists a concentrated set of reals of size \(\mathfrak{c} = \omega_2\). Let us recall that a set \(X\subseteq 2^\omega\) is concentrated if there exists a countable \(Q\subseteq X\) such that \(|X\setminus U|\leq \omega\) for every open set \(U \subseteq 2^\omega\) with \(Q\subseteq U\).

In our talk we shall present the main ideas of the proof that this conjecture is false. Concentrated sets are canonical examples of Rothberger spaces of reals. We want to analyse the possible cardinalities of sets of reals satisfying selection principles in the Miller model. To avoid triviality we are interested in the totally imperfect cases, i.e. spaces that do not contain a copy of the Cantor space. Note that since \(\mathfrak{d}\)-concentrated sets are totally imperfect Menger spaces, there are such spaces of size continuum (since \(\mathfrak{d} = \mathfrak{c}\)). We shall sketch the proof that for the strongest selection principle, the \(\gamma\)-set  property, only cardinality atmost \(\omega_1\) is possible. We hope that the tools of our results can be used as a prototype for the non-existence of Rothberger sets of reals with cardinality \(\mathfrak{c}\). The goal would be to prove the same for Hurewicz totally imperfect sets of reals, the latter being a weaker property than Rothberger in the Miller model.

The talk will be based on a recent joint work  with Piotr Szewczak and Lyubomyr Zdomskyy.