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Chapter IX of Proper and Improper Forcing addresses this issue. Shelah proves that Souslin's Hypothesis does not imply every Aronszajn tree is special, and he does this by investigating weak notions …

This is false in general when $\kappa=\omega$. Let $\mathcal{A}=\langle A_\alpha:\alpha<\mathfrak{c}\rangle$ be an almost disjoint family of size continuum, and let $\langle D_\alpha:\alpha<\mathfrak …

This is a question that connects to many things in set theory (and they are sometimes called ``strongly almost disjoint families"). First, an old result of Baumgartner (see Section 6 of [1]) shows th …

There IS an easy proof of this, but I just had to reframe the way I was thinking of the problem. The cardinals in question (and many of their relatives) turn out to be $2^{\mu}$ if $\mu$ is strong lim …

Assuming large cardinals, the answer is no even for $\aleph_\omega$. Given a supercompact cardinal, Ben-David and Magidor constructed a model in which $\aleph_{\omega+1}$ carries a uniform indecompos …

Assuming I understand the definitions correctly, I can give you a couple of references. (1) Dordal (see below) gives a model in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and all towers have cardinal …

Shelah has also considered this question in his paper [Sh:1008]. The published version indicates that he investigated this from scratch rather than starting with the Sharon-Viale observation on the Ab …