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Assume $V=L$ and let $\kappa$ be a weakly compact cardinal. Let $G$ be $Col(\aleph_1, < \kappa)$ generic over $V$. Working in $V[G]$ force with a countable support iteration of forcing notions of length $\kappa^+,$ where at each stage $\alpha,$ one takes a name $\dot{T}_\alpha$ for a $\kappa$-Souslin tree and forces to add an antichain into it of size $\kappa$ using countable approximations (one may assume that each level of the tree has size $\aleph_1$, so the forcing is non-trivial).

By Laver-Shelah, the forcing is $\kappa$-c.c. and by a suitable book-keeping argument, in the final extension there are no $\kappa=\aleph_2$-Souslin trees.

Question. Are there any non-special $\kappa$-Aronszajn trees in the above extension?

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  • $\begingroup$ Sounds like you're looking for a coauthor... :-) $\endgroup$ – Asaf Karagila Jul 12 '18 at 7:31

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