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.[-]

The purpose of this article is to give new constructions of linear orders which are minimal with respect to being non-σ-scattered. Specifically, we will show that Jensen's principle ♢ implies that there is a minimal Countryman line, answering a question of Baumgartner. We also produce the first consistent examples of minimal non-σ-scattered linear orders of cardinality greater than ℵ1, as given a successor cardinal κ+, we obtain such linear orderings of cardinality κ+ with the additional property that their square is the union of κ-many chains. We give two constructions: directly building such examples using forcing, and also deriving their existence from combinatorial principles. The latter approach shows that such minimal non-σ-scattered linear orders of cardinality κ+ exist for every cardinal κ in Gödel's constructible universe, and also (using work of Rinot) that examples must exist at successors of singular strong limit cardinals in the absence of inner models satisfying the existence of a measurable cardinal μ of Mitchell order μ++. [-]