No covering with nowhere dense \textsf{P}-sets in the Cohen model

Abstract

We prove that if less than $\aleph_{\omega}$-many Cohen reals are added to a model of \textsf{CH}, then $\omega^{\ast}$ can not be covered by nowhere dense \textsf{P}-sets (equivalently, there is an ultrafilter on $\omega$ that does not contain a tower).