Acta Mathematica Hungarica

, Volume 144, Issue 2, pp 285–352

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

• A. M. Brodsky
Article

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

## Key words and phrases

nonspecial tree stationary tree stationary subtree partial order diagonal union regressive function normal ideal Pressing–Down Lemma partition relation partition calculus Erdös–Rado Theorem Baumgartner–Hajnal–Todorcevic Theorem elementary submodel non-reflecting ideal very nice collection

## Mathematics Subject Classification

primary 03E02 secondary 03C62 05C05 05D10 06A07

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