More notions of forcing add a square

Joint work with Yair Hayut and Zhixing You.

Abstract.  Foreman and Magidor showed that the continuum hypothesis implies the existence of a countably-closed $\aleph_2$-cc forcing notion $\mathbb P$ for adding $\square_{\omega_1}$. Here, we show that $\mathbb P$ may consistently be realized as an $\aleph_2$-Souslin tree. More generally, we prove 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 singular cardinals. Our construction is uniform and extends to inaccessible cardinals as well.

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