Baumgartner's Axiom and Small Posets

TL;DR

This paper explores the consistency of certain dense sets of reals under Martin's Axiom, showing that specific structural properties can be preserved across forcing extensions, and provides new proofs for existing theorems.

Contribution

It demonstrates the consistency of $ extsf{MA}$ with the existence of $eth_1$-dense sets of reals having particular non-isomorphic uncountable subsets in forcing extensions, offering new insights and proofs.

Findings

Existence of $eth_1$-dense sets of reals consistent with $ extsf{MA}$

Such sets do not contain uncountable order-isomorphic subsets in certain extensions

Provides alternative proof of a theorem by Moore and Todorcevic

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 $\textsf{MA}$ 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 $A^*$ 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.