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.