HUJI Logic Seminar
Speaker: Chris Lambie-Hanson
Title: Reflections on the coloring and chromatic numbers
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.