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Question 1. What is known about the consistency strength of $\aleph_2$-Souslin hypothesis?

Question 2. What is known about the consistency strength of having both $\aleph_2$-Souslin hypotheis and $\aleph_3$-Souslin hypothesis?

Remark 1. By $\kappa$-Souslin hopothesis, I mean there are no $\kappa$-Souslin trees.

Remark 2. By Laver-Shelah, the existence of a weakly compact cardinal implies the consistency of $\aleph_2$-Souslin hypothesis. On the other hand by results of Shelah-Stanly, if we assume some instances of $GCH$+ $\aleph_2$-Souslin hypothesis (having $CH$ is sufficient), then some large cardinals (at least Mahlo) are required. In the above question I do not take care of preserving instances of $GCH$.

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Answer to 1, without CH:

  • Mitchell and Silver, 1973: Weakly compact is an upper bound.

Answer to 1, with CH:

  • Laver and Shelah, 1981: Weakly compact is an upper bound.
  • Shelah and Stanley, 1982: Inaccessible is a lower bound.

Answer to 1, with GCH:

  • Gregory, 1976: Mahlo cardinal is a lower bound.
  • Rinot, 2016: Weakly compact is a lower bound.

Answer to 2, with GCH:

  • Rinot, 2016 (building on recent work of Schindler and Steel): AD holds in $L(\mathbb R)$ is a lower bound.
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  • $\begingroup$ my paper is available here: assafrinot.com/paper/24 $\endgroup$
    – saf
    Commented Apr 4, 2016 at 15:59
  • $\begingroup$ Very nice Assaf. I'm still wondering what happens if we assume no $GCH$ type assumptions. $\endgroup$
    – Mohammad Golshani
    Commented Apr 5, 2016 at 4:02

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