We construct a model in which MA[Formula: see text] holds and [Formula: see text] fails. This shows that MA[Formula: see text] does not imply [Formula: see text] and answers an old question of Larson and Todorcevic in [Katetov’s problem, Trans. Amer. Math. Soc. 354(5) (2002) 1783–1791]. We also investigate different strong colorings in models of MA[Formula: see text].

The Axiom of Strong Condensation, first introduced by Woodin in [14], is an abstract version of the Condensation Lemma ofL. In this paper, we construct a set-sized forcing to obtain Strong Condensation forH. As an application, we show that “ZFC + Axiom of Strong Condensation +”is consistent, which answers a question in [14]. As another application, we give a partial answer to a question of Jech by proving that “ZFC + there is a supercompact cardinal + any ideal onω1which is (...) definable overH is not precipitous” is consistent under sufficient large cardinal assumptions. (shrink)