We show that, consistently, for some regular cardinals θ <λ, there exists a Boolean algebra 𝔹 such that |𝔹| = λ⁺ and for every subalgebra 𝔹'⊆ 𝔹 of size λ⁺ we have Depth(𝔹') = θ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-1-7, author = {Andrzej Ros\l anowski and Saharon Shelah}, title = {Historic forcing for Depth}, journal = {Colloquium Mathematicae}, volume = {89}, year = {2001}, pages = {99-115}, zbl = {0982.03029}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-1-7} }
Andrzej Rosłanowski; Saharon Shelah. Historic forcing for Depth. Colloquium Mathematicae, Tome 89 (2001) pp. 99-115. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-1-7/