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!

  • 3
    $\begingroup$ Terminological disaster: The guessing principle named "club" is not the same thing as club-guessing. $\endgroup$ Jun 4, 2015 at 19:34
  • 2
    $\begingroup$ Fortunately, there is a friendly disambiguating ambulance in the vicinity: papers.assafrinot.com/the_search_for_diamonds.pdf $\endgroup$
    – Avshalom
    Jun 4, 2015 at 23:37

1 Answer 1


The answer is negative apparently. It is consistent relative to ZFC that all Aronszajn trees are special and that the club principle holds:


  • $\begingroup$ The link is broken. $\endgroup$ May 26, 2019 at 14:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.