Forcing Over a Free Suslin Tree
John Krueger, Sarka Stejskalova
TL;DR
The paper develops an abstract forcing framework for forcing over a free Suslin tree, then applies it to three constructs: specializing derived trees, adding almost disjoint subtrees, and adding automorphisms. Each application builds a tailored forcing (with countable conditions and Key Properties) that preserves Suslinness and avoids introducing new cofinal branches, while enforcing the desired structure on the tree. The main achievement is a model, under an inaccessible κ, where Col$(\omega_1,<\kappa)$ combined with the automorphism-based forcing produces a normal Suslin tree with an ω$_2$-sized almost disjoint automorphism family and no Kurepa tree, yielding an almost Kurepa Suslin tree and a non-saturated Aronszajn tree. These results resolve several open questions about the existence/absence of Kurepa trees, almost Kurepa trees, and rigidity phenomena in Suslin trees. Overall, the work provides a cohesive framework linking consistency, separation, and the Key Property to powerful forcing constructions with precise preservation properties.
Abstract
We introduce an abstract framework for forcing over a free Suslin tree with suborders of products of forcings which add some structure to the tree using countable approximations. The main ideas of this framework are consistency, separation, and the Key Property. We give three applications of this framework: specializing derived trees of a free Suslin tree, adding uncountable almost disjoint subtrees of a free Suslin tree, and adding almost disjoint automorphisms of a free Suslin tree. Using the automorphism forcing, we construct a model in which there is an almost Kurepa Suslin tree and a non-saturated Aronszajn tree, and there does not exist a Kurepa tree. This model solves open problems due to Bilaniuk, Moore, and Jin and Shelah.