Universally meager sets in the Miller model and similar ones

Abstract

We work in the realm of sets of reals. We prove that in the Miller model and in a model constructed by Goldstern-Judah-Shelah all universally meager sets have size at most $\omega_1$. Some relations between combinatorial covering properties in these models allow to obtain the same limitations for sizes of Rothberger spaces and Hurewicz spaces with no homeomorphic copy of the Cantor set inside. It follows from our results that the existence of a strong measure zero set of size $\omega_2$ does not imply the existence of a Rothberger space of size $\omega_2$. We also prove that in the Miller model all strong measure zero sets have size at most $\omega_1$.