A model with Suslin trees but no minimal uncountable linear orders other than $\omega_1$ and $-\omega_1$

TL;DR

The paper demonstrates that having a Suslin tree does not imply the existence of uncountable minimal linear orders beyond the well-known  and , using a specialized forcing technique.

Contribution

It introduces a Jensen-type iteration to show the consistency of CH with a restricted ladder system uniformization on trees, preserving a Suslin tree without additional minimal linear orders.

Findings

Suslin trees can exist without uncountable minimal linear orders other than  and .

Forcing can produce models with CH and specific ladder system properties while maintaining Suslin trees.

The results answer a question posed by J. Baumgartner.

Abstract

We show that the existence of a Suslin tree does not necessarily imply that there are uncountable minimal linear orders other than $\omega_1$ and $-\omega_1$, answering a question of J. Baumgartner. This is done by a Jensen-type iteration, proving that one can force CH together with a restricted form of ladder system uniformization on trees, all while preserving a rigid Suslin tree.