Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees

Building on early work by Stevo Todorcevic, we describe a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem: Let $\kappa$ be any infinite regular cardinal, let $\xi$ be any ordinal such that $2^{\left|\xi\right|} < \kappa$, and let $k$ be any natural number. Then $$\text{non-$\left(2^{<\kappa}\right)$-special tree } \to \left(\kappa + \xi \right)^2_k.$$ This is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above to the cardinal $(2^{<\kappa})^+$, the simplest example of a non-$(2^{<\kappa})$-special tree. As a corollary, we obtain a general result for partially ordered sets: Theorem: Let $\kappa$ be any infinite regular cardinal, let $\xi$ be any ordinal such that $2^{\left|\xi\right|} < \kappa$, and let $k$ be any natural number. Let $P$ be a partially ordered set such that $P \to (2^{<\kappa})^1_{2^{<\kappa}}$. Then $$P \to \left(\kappa + \xi \right)^2_k.$$Comment: Submitted to Acta Mathematica Hungaric

The non-absoluteness of model existence in uncountable cardinals for Lw1,w

"Vegeu el resum a l'inici del document del fitxer adjunt"

Square compactness and Lindel\"of trees

We prove that every weakly square compact cardinal is a strong limit cardinal. We also study Aronszajn trees with no uncountable finitely branching subtrees, characterizing them in terms of being Lindel\"of with respect to a particular topology. We prove that the class of such trees lies between the classes of Suslin and Aronszajn trees, and that the inclusions can consistently be strict

Set Theory

This workshop included selected talks on pure set theory and its applications, simultaneously showing diversity and coherence of the subject