($\lozenge$) There exists Suslin trees such that their product is Suslin too.

Question

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.

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.