Abstract: The notion of an ascent path through a tree, isolated by Laver, is a generalization of the notion of a cofinal branch and, in many cases, the existence of an ascent path through a tree provides a concrete obstruction to the tree being special. We will discuss some recent results regarding ascent paths through kappa-trees, where kappa > omega_1 is a regular cardinal. We will discuss the consistency of the existence or non-existence of a special mu^+-tree with a cf(mu)-ascent path, where mu is a singular cardinal. We will also discuss the consistency of the statement, "There are omega_2-Aronszajn trees but every omega_2-tree contains an omega-ascent path." We will connect these topics with various square principles and with results about the productivity of chain conditions.
Wed, 22/03/2017 - 16:00 to 18:00