Perfect Subtree Property for Weakly Compact Cardinals

(with Y. Hayut)

Submitted. PDF. arXiv. Bibtex.

In this paper we investigate the consistency strength of the statement: $\kappa$ is weakly compact and there is no tree on $\kappa$ with exactly $\kappa^{+}$ many branches. We show that this property fails strongly (there is a sealed tree) if there is no inner model with a Woodin cardinal. On the other hand, we show that this property as well as the related Perfect Subtree Property for $\kappa$, implies the consistency of $\operatorname{AD}_{\mathbb{R}}$.