### 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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