Specializing trees and answer to a question of Williams

Abstract

We show that if [Formula: see text] then any nontrivial [Formula: see text]-closed forcing notion of size [Formula: see text] is forcing equivalent to [Formula: see text] the Cohen forcing for adding a new Cohen subset of [Formula: see text] We also produce, relative to the existence of suitable large cardinals, a model of [Formula: see text] in which [Formula: see text] and all [Formula: see text]-closed forcing notion of size [Formula: see text] collapse [Formula: see text] and hence are forcing equivalent to [Formula: see text] These results answer a question of Scott Williams from 1978. We also extend a result of Todorcevic and Foreman–Magidor–Shelah by showing that it is consistent that every partial order which adds a new subset of [Formula: see text] collapses [Formula: see text] or [Formula: see text]