Open Colorings and Baumgartner's Axiom

Abstract

We construct a model of $\mathsf{M}{\mathsf{A}}_{{\aleph }_1}+{\mathsf{OCA}}_T$ where Baumgartner's Axiom fails, settling a question of Farah. Moreover, in the same model there is an ${\aleph }_1$-dense set of reals which is neither reversible nor increasing, answering a question of Marun, Shelah, and Switzer.