## 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 |
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Pages (from-to) | 285-352 |

Number of pages | 68 |

Journal | Acta Mathematica Hungarica |

Volume | 144 |

Issue number | 2 |

DOIs | |

State | Published - 1 Nov 2014 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2014, Akadémiai Kiadó, Budapest, Hungary.

## Keywords

- Baumgartner–Hajnal–Todorcevic Theorem
- Erdös–Rado 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