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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!

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

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  • $\begingroup$ The link is broken. $\endgroup$ – Ari Brodsky May 26 at 14:25

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