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

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].