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 languageEnglish
Pages (from-to)285-352
Number of pages68
JournalActa Mathematica Hungarica
Volume144
Issue number2
DOIs
StatePublished - 1 Nov 2014
Externally publishedYes

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

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