I'm trying to show that under $\lozenge$ assumption there exists $S$ and $T$ Suslin trees such that $S\otimes T$ is also Suslin. I really have no idea how to use the existence of a $\lozenge$-sequence.

I'm looking just for a hint, not an answer.

Thank you.

  • $\begingroup$ This is theorem 6.6 (page 268) in Todorcevic's chapter in Handbook of set-theoretic topology. Try looking at that construction. $\endgroup$
    – hot_queen
    Sep 16, 2015 at 22:52
  • 1
    $\begingroup$ Another reference treating this and many variants in detail is the Abraham-Shelah paper A $\Delta^2_2$ well-order of the reals and incompactness of $L(Q^{\rm MM})$, Ann. Pure Appl. Logic, 59 (1), (1993), 1–32. $\endgroup$ Sep 17, 2015 at 13:03

1 Answer 1


Probably too late, but still a hint: As the $\lozenge$-sequence consists of subsets of $\omega_1$, use it to build a "$\lozenge$-sequence" of subsets of $\omega_1\times\omega_1$, and then use that sequence to build the three trees $T,S,T\otimes S$ simultanousely, immitating the construction of one tree.


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