Baumgartner's Axiom and Small Posets

Abstract

We contribute to the study of ${\aleph }_1$-dense sets of reals, a mainstay in set theoretic research since Baumgartner's seminal work in the 70s. In particular, we show that it is consistent with $\text{MA}$ that there exists an ${\aleph }_1$-dense set of reals $A$ so that, in any cardinal-preserving generic extension by a forcing of size ${\aleph }_1$, $A$ and $A^{\ast }$ do not contain uncountable subsets which are order isomorphic. This strengthens a result of Avraham and the second author and yields a different proof of a theorem of Moore and Todorcevic.