A celebrated theorem of Shelah states that adding a Cohen real introduces a Souslin tree. Are there any other examples of notions of forcing that add a $\kappa$-Souslin tree? and why is this of interest?

My motivation comes from a question of Schimmerling, which I shall now motivate and state.

Recall that Jensen proved that GCH together with the square principle $\square_\lambda$ entails a $\lambda^+$-Souslin tree for all cardinals $\lambda\ge\aleph_1$. Recently, it was shown that $\square_\lambda$ may be replaced by the weaker principle $\square(\lambda^+)$. Of course, another weakening of $\square_\lambda$ is the principle $\square^*_\lambda$.

Schimmerling’s question indeed asks whether it is consistent with GCH that $\square^*_\lambda$ holds for a singular cardinal $\lambda$, and yet there exist no $\lambda^+$-Souslin trees.

The first line of attacks that comes to mind here would involve Prikry/Magidor/Radin forcing to singularize a former large cardinal (e.g., this paper).

In this post, we announce a (corollary to a) theorem from an upcoming paper with Brodsky, showing that this line of attacks is a no-go.

**Theorem.** Suppose that $\lambda$ is a strongly inaccessible cardinal satisfying $2^\lambda=\lambda^+$. If $\mathbb P$ is a $\lambda^+$-cc notion of forcing of size $\le\lambda^+$ that singularizes $\lambda$, then $\mathbb P$ adds a $\lambda^+$-Souslin tree.

I think that this adds nicely to the paper Yair and I wrote. Forcings that singularize cardinals do nasty things.

Souslin trees are of diamond-quality; quite far from being considered nasty here :).

Well… not if you are trying to prove the consistency of weak squares without Suslin trees at a singular cardinal. 🙂

That would be quite a depressing model.

In continuation of my recent post about my experience of Norwich, this conversation should continue only after the third beer has been served.

😛

Do you know if this is also true for Prikry or Radin forcing if $2^\lambda >\lambda^+$? This is the situation in the Cummings, et al. paper you linked to and seems like it would naturally arise in attempts to answer Schimmerling’s question in a similar way.

Indeed, that’s the situation in the Cummings et al paper, but Schimmerling asks about GCH. About our argument: all of the Souslin trees constructions in the “Microscopic approach” papers are $diamondsuit$-based.

Once we are done with this project, I would definitely like to think about constructions that are not $diamondsuit$-based.

Right, I missed the inclusion of GCH in Schimmerling’s question. As far as I know, it’s already an interesting and open question whether there can be a singular cardinal $\lambda$ such that the tree property fails at $\lambda^+$ and yet there are no $lambda^+$-Souslin trees, regardless of cardinal arithmetic assumptions.

I agree!

Does your result says anything when $2^\lambda > \lambda^+$?

It does not, because the proof goes through the proxy principle $P(\lambda^+,\cdots)$ that asserts, among other things, that $\diamondsuit(\lambda^+)$ holds.