## Finitely branching subtrees of Aronszajn trees

Young Set Theory Workshop 2023, University of Münster, Münster, Germany

## Subtrees with small branching number

2023 ASL North American Annual meeting, UC Irvine, Irvine, California

Given a tree $T$ , we can view a branch through $T$ as a subtree which is $1$-branching, i.e. each point on the branch has exactly one immediate successor on the branch. This point of view allows us to generalize the tree property by asking whether a tree can have tall subtrees with small branching number. In particular, we can ask whether an Aronszajn tree can have finitely branching subtrees of height $\aleph_1$. This leads to a class of trees which lies between those of Suslin and Aronszajn trees. In addition, such trees can be characterized as those having the Lindelöf property with respect to a reasonably nice topology.

## Square compactness and dense linear orders

New Frontiers in Set Theoretic Topology Workshop, University of Pittsburgh, Pittsburgh, Pennsylvania

Unlike compactness, the Lindelöf property need not be preserved under products. The analogous properties at bigger cardinals leads one to large cardinal notions. We extend the classical Sorgenfrey plane construction to larger spaces, provided some cardinal arithmetic assumptions hold. This allows us to remove a hypothesis from a result of Buhagiar and Džamonja.