This is problem 15.3 in Arnie Miller's [problem list](http://www.math.wisc.edu/~miller/res/problems.pdf): 


(Juhasz) Suppose there exists $\langle A\_{\alpha} : \alpha \in L \rangle$, where $L$ is the set of limit ordinals below $\omega\_1$ and for each $\alpha \in L$, $A\_{\alpha}$ is an unbounded subset of $\alpha$, satisfying: For every unbounded $A \subseteq \omega\_1$, there exists $\alpha \in L$, $A\_{\alpha} \subseteq A$. Must there exists a Suslin tree?


What is the current status of this problem? Thanks!



Is $\clubsuit_{\omega_1}$ enough to get Suslin tree?
The answer is negative apparently. It is consistent relative to ZFC that all Aronszajn trees are
special and that the club principle holds: 


<http://home.mathematik.uni-freiburg.de/mildenberger/postings/paperspdf/988_2014_10_15no.pdf>


