Martin’s axiom has been very successful in deciding numerous questions such as the Souslin hypothesis. Can we find higher analogues that, for example, decide the $\kappa$-Souslin hypothesis and allow us to meet $< 2^\kappa$ dense open sets? If one is careful enough and imposes closure conditions and stronger forms of the $\kappa$-cc, one obtains the principle BA$_\kappa$, independently discovered by Baumgartner, Laver and Shelah. BA$_\kappa$ shares many similarities with Martin’s axiom but decides the $\kappa$-Souslin hypothesis in the wrong way.