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 ξ be any ordinal such that ${2^{|\xi|} < \kappa}$, and let k be any natural number. Then

$$non-(2^{<\kappa})-special\, tree \rightarrow (\kappa + \xi)^{2}_k.$$

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 ξ be any ordinal such that ${2^{|\xi|} < \kappa}$, and let k be any natural number. Let P be a partially ordered set such that ${P \rightarrow (2^{< \kappa})^{1}_{2^{< \kappa}} }$. Then

$$P \rightarrow (\kappa + \xi)^{2}_{k}.$$