MA$_{\omega_1}(S)[S]$ does not imply $\mathcal{K}_2$

TL;DR

The paper constructs a model demonstrating that MA$_{oldsymbol{ ext{$oldsymbol{ ext{MA}}_{oldsymbol{ ext{$oldsymbol{ ext{$ ext{omega}_1}$}}}$}(S)[S]}$} and $oldsymbol{ ext{ ext{$ ext{K}_2$}}}$ are independent, answering a longstanding question in set theory.

Contribution

It provides a model where MA$_{ ext{omega}_1}$(S)[S] holds but $ ext{K}_2$ fails, showing these principles are independent.

Findings

MA$_{ ext{omega}_1}$(S)[S] does not imply $ ext{K}_2$

Different strong colorings are analyzed in models of MA$_{ ext{omega}_1}$(S)[S]

Answers an old question of Larson and Todorcevic

Abstract

We construct a model in which MA$_{\omega_1}$(S)[S] holds and $\mathcal{K}_2$ fails. This shows that MA$_{\omega_1}$(S)[S] does not imply $\mathcal{K}_2$ and answers an old question of Larson and Todorcevic in [3]. We also investigate different strong colorings in models of MA$_{\omega_1}$(S)[S].