Universally meager sets in the Miller model and similar ones

TL;DR

This paper investigates the size limitations of universally meager sets, Rothberger spaces, and Hurewicz spaces in the Miller model and similar models, revealing new bounds and relationships among these sets.

Contribution

It establishes that in the Miller and related models, universally meager and strong measure zero sets are at most size , and explores implications for Rothberger and Hurewicz spaces.

Findings

Universally meager sets have size at most  in the Miller model.

Strong measure zero sets are at most size  in the Miller model.

Existence of -sized strong measure zero sets does not imply -sized Rothberger spaces.

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 $\omega_1$.