Specializing trees and answer to a question of Williams

by Golshani and Shelah. [GsSh:1120]

We show that if cf(2^{aleph_0}) = aleph_1, then any non-trivial aleph_1-closed forcing notion of size <= 2^{aleph_0} is forcing equivalent to Add(aleph_1, 1), the Cohen forcing for adding a new Cohen subset of omega_1 . We also produce, relative the existence of some large cardinals, a model of ZFC in which 2^{aleph_0} = aleph_2 and all aleph_1-closed forcing notion of size <= 2^{aleph_0} collapse aleph_2, and hence are forcing equivalent to Add(aleph_1, 1) . Our 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 aleph_2, collapes aleph_2 or aleph_3 .

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