# Ari Meir Brodsky: A theory of non-special trees, and a generalization of the Balanced Baumgartner-Hajnal-Todorcevic Theorem

12/July/2013, 13:30-15:00
Fields Institute, Room 210

Speaker:  Ari Meir Brodsky

Title:  A theory of non-special trees, and a generalization of the Balanced Baumgartner-Hajnal-Todorcevic Theorem

Abstract:

Building on early work by Stevo Todorcevic, we describe a theory of non-special 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 tree as being either stationary or non-stationary. We then use this theory to prove a partition relation for trees:

THEOREM: Let $\nu$ and $\kappa$ be cardinals such that $\nu ^ {<\kappa} = \nu$, and let $T$ be a non-special tree of height $\nu^+$. Then for any ordinal $\xi$ such that $2^{\left|\xi\right|} < \kappa$, and finite $k$, we have $T \to (\kappa + \xi )^2_k$.

This is a generalization of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which is the special case of the above where the tree $T$ is replaced by the cardinal $\nu^+$.