Israel Journal of Mathematics. Volume 253, March 2023. Pages 865-886.
DOI: 10.1007/s11856-022-2385-4. PDF. arXiv. Bibtex.

 @ARTICLE{HM19,
   AUTHOR= {Y. Hayut and S. Müller},
   TITLE= {Perfect subtree property for weakly compact cardinals},
   JOURNAL={Israel Journal of Mathematics},
   VOLUME = {253},
   ISSUE = {},
   YEAR = {2023},
   PAGES = {865--886},
   EPRINT = {1910.05159},
   DOI = {10.1007/s11856-022-2385-4}}

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 $AD_{\mathbb{R}} + DC$.