Ultrafilters with property $( {s})$

Abstract:

A set $X\subseteq 2^\omega$ has property (s) (Marczewski (Szpilrajn)) iff for every perfect set $P\subseteq 2^\omega$ there exists a perfect set $Q\subseteq P$ such that $Q\subseteq X$ or $Q\cap X=\emptyset$. Suppose ${\mathcal {U}}$ is a nonprincipal ultrafilter on $\omega$. It is not difficult to see that if ${\mathcal {U}}$ is preserved by Sacks forcing, i.e., if it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then ${\mathcal {U}}$ has property (s) in the ground model. It is known that selective ultrafilters or even P-points are preserved by Sacks forcing. On the other hand (answering a question raised by Hrusak) we show that assuming CH (or more generally MA) there exists an ultrafilter ${\mathcal {U}}$ with property (s) such that ${\mathcal {U}}$ does not generate an ultrafilter in any extension which adds a new subset of $\omega$.