Journal of Mathematical Logic 21 (1):2050023 (2020)

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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]
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DOI 10.1142/s0219061320500233
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References found in this work BETA

Fragility and Indestructibility of the Tree Property.Spencer Unger - 2012 - Archive for Mathematical Logic 51 (5-6):635-645.
Morasses and the Lévy-Collapse.P. Komjáth - 1987 - Journal of Symbolic Logic 52 (1):111-115.
0♯ and Some Forcing Principles.Matthew Foreman, Menachem Magidor & Saharon Shelah - 1986 - Journal of Symbolic Logic 51 (1):39 - 46.
On Foreman’s Maximality Principle.Mohammad Golshani & Yair Hayut - 2016 - Journal of Symbolic Logic 81 (4):1344-1356.

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