More notions of forcing add a Souslin tree

TL;DR

This paper explores various forcing notions, including Cohen, Prikry, Magidor, and Radin, demonstrating that many add -Souslin trees under GCH assumptions, expanding understanding of their combinatorial properties.

Contribution

It identifies a broad class of forcing notions that add -Souslin trees, including well-known forcings like Prikry, Magidor, and Radin, under GCH assumptions.

Findings

Cohen forcing adds -Souslin trees.

Prikry, Magidor, and Radin forcing also add -Souslin trees.

The class of forcing notions that add -Souslin trees is large under GCH.

Abstract

An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion --- Cohen forcing --- adds an $\aleph_1$-Souslin tree. In this paper, we identify a rather large class of notions of forcing that, assuming a GCH-type assumption, add a $\lambda^+$-Souslin tree. This class includes Prikry, Magidor and Radin forcing.