More notions of forcing add square

Joint work with Yair Hayut and Zhixing You.

Abstract. Foreman and Magidor proved that the continuum hypothesis yields a countably-closed $\aleph_2$-cc notion of forcing $\mathbb P$ for adding $\square_{\omega_1}$. Here we prove that $\mathbb P$ may consistently be taken to be an $\aleph_2$-Souslin tree.
More generally, it is shown that $\square_\lambda$ may be added by a $\lambda^+$-Souslin tree, providing the first analog of the Foreman–Magidor forcing at the level of successors of singulars. Our construction is uniform and applies to inaccessible cardinals as well.

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