Recall Silver's theorem which says that "if GCH holds below $\aleph_{\omega_1}$, then $2^{\aleph_{\omega_1}}=\aleph_{\omega_1+1},$ i.e., it also holds at $\aleph_{\omega_1}$".

Recently, Gitik has proved a similar result for collapses. See his paper [Silver type theorems for collapses.
](http://www.math.tau.ac.il/~gitik/Silverforcollapses3.pdf)

Here I would like to ask two related questions:

>**Question 1.** Suppose that for each limit ordinal $\alpha < \omega_1,$ there exists a special $\aleph_{\alpha+1}$-Aronszajn tree. Does it follow that there is a special  $\aleph_{\omega_1+1}$-Aronszajn tree?

The next question is related to tree property.

>**Question 2.** Suppose that for each limit ordinal $\alpha < \omega_1,$ there exists an $\aleph_{\alpha+1}$-Aronszajn tree. Does it follow that there is an  $\aleph_{\omega_1+1}$-Aronszajn tree?

**Remark.** Let me say why a negative consistency result to, say question 2, is not at least trivial. The simplest way to get the tree property at successor of a singular cardinal $\kappa$ with, say, $cf(\kappa)=\omega_1,$ is to have   an increasing and cofinal sequence $(\kappa_\xi: \xi < \omega_1)$
of strongly compact cardinals below $\kappa$. But then, for each limit ordinal $\alpha < \omega_1,$ the tree property holds at the successor of $\sup_{\xi<\alpha}\kappa_\xi$ as well. So $\kappa^+$ in not the least successor of a singular cardinal for which the tree property holds. The natural idea is to kill all possible such cases below $\kappa,$ but then it is not clear to me if we preserve tree property at $\kappa^+.$ 

On the other hand, maybe the arguments similar to those given by Ari Meir Brodsky and Assaf Rinot in [Reduced powers of Souslin trees
](http://blog.assafrinot.com/?p=4007) may give a positive answer to the above questions.