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]

中文翻译：

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专业化树木并回答威廉姆斯的问题

我们证明如果 [Formula: see text] 那么任何非平凡的 [Formula: see text] - 封闭的大小 [Formula: see text] 强制概念等效于 [Formula: see text] 添加新 Cohen 子集的 Cohen 强制[公式：见文本]我们还产生了，相对于合适的大基数的存在，[公式：见文本]的模型，其中[公式：见文本]和所有[公式：见文本]-封闭强迫概念size [Formula: see text] collapse [Formula: see text] 因此强制等价于 [Formula: see text] 这些结果回答了 Scott Williams 从 1978 年开始提出的问题。我们还扩展了 Todorcevic 和 Foreman–Magidor–Shelah 的结果通过证明添加[公式：参见文本]的新子集的每个偏序折叠[公式：参见文本]或[公式：参见文本]是一致的