On the rigidity of Souslin trees and their generic branches

TL;DR

This paper explores the rigidity properties of Souslin trees under forcing, demonstrating the consistency of certain rigidity and Kurepa properties, and addressing questions about dense subsets and structural uniqueness.

Contribution

It establishes the consistency of Souslin trees being Kurepa after forcing and introduces new results on the existence of dense subsets without copies, answering open questions.

Findings

Existence of a Souslin tree that becomes Kurepa after forcing

For every Souslin tree, there exists a dense subset without a copy of the original

Consistency results related to rigidity and dense subsets in Souslin trees

Abstract

We show it is consistent that there is a Souslin tree $S$ such that after forcing with $S$, $S$ is Kurepa and for all clubs $C \subset \omega_1$, $S\upharpoonright C$ is rigid. This answers Fuchs's questions in Club degrees of rigidity and almost Kurepa trees. Moreover, we show it is consistent with $\diamondsuit$ that for every Souslin tree there is a dense $X \subset S$ which does not have a copy of $S$. This is related to a question due to Baumgartner.