By a space we mean a metrizable separable zero-dimensional space. A space $X\subseteq 2^{\omega }$ is universally meager if for any Polish space $Y$ and any continuous nowhere constant map $f:Y\to 2^{\omega }$ the preimage $f^{-1}[X]$ is meager in $Y$. We call a space totally imperfect if it contains no copy of $2^{\omega }$. We present a forcing property $(†)$, which is a strenthening of properness and implies that no dominating reals are added. It is known that many classical forcing posets like Cohen, Sacks and Miller satisfy this property. We showed that property $(†)$ is preserved by countable support iterations.

We then used this preservation result to prove that if we have such an iteration of length ${\omega }_2$ over a model of CH, where the single forcings have size at most ${\omega }_1$, all universally meager sets $X\subseteq 2^{\omega }$ have size at most ${\omega }_1$ in the forcing extension.
This has multiple set-theoretic applications:

In the Miller model, we generalized our result of having no concentrated and $\gamma$-sets of size continuum to totally imperfect Hurewicz sets, which are universally meager by a result of Zakrzewski. Moreover, since Bartoszyński showed that all perfectly meager spaces are universally meager in the Miller model, we get that indeed even all perfectly meager spaces have size stricly less than continuum in the Miller model.

Miller proved in 2005 that there exists a strong measure zero set of size ${\omega }_1$ iff there exists a Rothberger space of size ${\omega }_1$. Goldstern, Judah and Shelah constructed in 1993 a forcing iteration for which there is a strong measure zero set of size ${\omega }_2$ in the extension. However, this iteration satisfies property $(†)$ and Rothberger spaces are universally meager in this model. Hence our result implies that it is consistent with ZFC to have a strong measure zero set of size ${\omega }_2$, but no Rothberger spaces of size ${\omega }_2$.

This is joint work with Piotr Szewczak (Cardinal Stefan Wyszyński University in Warsaw) and Lyubomyr Zdomskyy (TU Vienna).