Ultrafilters with property $( {s})$

Abstract

A set X ⊆ 2 ! has property (s) (Marczewski (Szpilrajn)) iff for every perfect set P ⊆ 2 ! there exists a perfect set Q ⊆ P such that Q ⊆ X or Q∩X = ∅. Suppose U is a nonprincipal ultrafilter on !. It is not difficult to see that ifU is preserved by Sacks forcing, i.e., it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then 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 U with property (s) such that U does not generate an ultrafilter in any extension which adds a new subset of !.