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

A. M. Brodsky

Research output: Contribution to journalArticlepeer-review

## 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 k be any infinite regular cardinal, letξ be any ordinal such that (Formula presented.), and let k be any natural number. Then (Formula presented.) This is a generalization to trees of the Balanced Baumgartner–Hajnal–Todorcevic Theorem, which we recover by applying the above to the cardinal (Formula presented.), the simplest example of a non-(Formula presented.)-special tree.

As a corollary, we obtain a general result for partially ordered sets.

Theorem. Let k (Formula presented.) be any infinite regular cardinal, letξ be any ordinal such that (Formula presented.), and let k be any natural number. LetPbe a partially ordered set such that(Formula presented.). Then(Formula presented.).

Original language English 285-352 68 Acta Mathematica Hungarica 144 2 https://doi.org/10.1007/s10474-014-0419-z Published - 1 Nov 2014 Yes

## Keywords

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

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