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

Abstract

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