Perfect Subtree Property for Weakly Compact Cardinals

Abstract

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 statement fails strongly (in the sense that there is a sealed tree with exactly ${\kappa }^{+}$ many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact cardinal $\kappa$ the nonexistence of a tree on $\kappa$ with exactly ${\kappa }^{+}$ many branches and, in particular, the Perfect Subtree Property for $\kappa$, implies the consistency of $A{D}_{\mathbb{R}}+DC$.