# Is $\clubsuit_{\omega_1}$ enough to get Suslin tree?

This is problem 15.3 in Arnie Miller's problem list:

(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!

• Terminological disaster: The guessing principle named "club" is not the same thing as club-guessing. – Andreas Blass Jun 4 '15 at 19:34
• Fortunately, there is a friendly disambiguating ambulance in the vicinity: papers.assafrinot.com/the_search_for_diamonds.pdf – Avshalom Jun 4 '15 at 23:37