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

Infinite Combinatorics Seminar (BIU)

23/November/2014, 10:15-12:00,
Room 201, Building 216, Bar-Ilan University

Speaker: Ari Meir Brodsky (BIU)

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

Abstract: Building on early work by Stevo Todorcevic, we develop 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 $P \to \left(\kappa + \xi \right)^2_k$.

A full exposition of the results is contained in my PhD thesis, available here.

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