# A forcing axiom deciding the generalized Souslin Hypothesis

Joint work with Chris Lambie-Hanson.

Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees.
It follows that for every uncountable cardinal $\lambda$, if $\lambda^{++}$ is not a Mahlo cardinal in Godel’s constructible universe, then $2^\lambda = \lambda^+$ entails the existence of a  $\lambda^+$-complete $\lambda^{++}$-Souslin tree.