Open Colorings and Baumgartner's Axiom

TL;DR

This paper constructs a model demonstrating the independence of Baumgartner's Axiom from certain set-theoretic assumptions, and shows the existence of a special dense set of reals with unique properties.

Contribution

It provides a new model where Baumgartner's Axiom fails alongside Martin's Axiom and the Open Coloring Axiom, answering longstanding open questions.

Findings

Baumgartner's Axiom can fail under MA + OCA_T.

Existence of a non-reversible, non-increasing -dense set of reals.

The model settles questions posed by Farah, Marun, Shelah, and Switzer.

Abstract

We construct a model of $\mathsf{MA_{\aleph_1}}+\mathsf{OCA}_T$ 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.