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

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