By the Cantor-Bendixson theorem, subtrees of the binary tree on $\omega$ satisfy a dichotomy - either the tree has countably many branches or there is a perfect subtree (and in particular, the tree has continuum manybranches, regardless of the size of the continuum). We generalize this to arbitrary regular cardinals $\kappa$ and ask whether every $\kappa$-tree with more than $\kappa$ branches has a perfect subset. From large cardinals, this statement isconsistent at a weakly compact cardinal $\kappa$. We show using stacking mice that the existence of a non-domestic mouse (which yields a model with a proper class of Woodin cardinals and strong cardinals) is a lower bound. Moreover, we study variants of this statement involving sealed trees, i.e. trees with the property that their set of branches cannot be changed by certain forcings, and obtain lower bounds for these as well. This is joint work with Yair Hayut.[-]

Universally Baire sets play an important role in studying canonical models with large cardinals. But to reach higher large cardinals more complicated objects, for example, canonical subsets of universally Baire sets come into play. Inspired by core model induction, we introduce the definable powerset $A^{\infty }$ of the universally Baire sets $\Gamma ^{\infty }$ and show that, after collapsing a large cardinal, $L(A^{\infty })$ is a model of determinacy and its theory cannot be changed by forcing. Moreover, we show a similar result for adding a club filter to the model constructed over universally Baire sets.[-]