A THEORY OF STATIONARY TREES AND THE BALANCED BAUMGARTNER-HAJNAL-TODORCEVIC THEOREM FOR TREES

作者：Brodsky A M^*

来源：Acta Mathematica Hungarica, 2014, 144(2): 285-352.

DOI：10.1007/s10474-014-0419-z

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

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 xi be any ordinal such that 2(vertical bar xi vertical bar) < kappa, and let kappa be any natural number. Then
non-(2(<kappa))-special tree -> (kappa + xi)(k)(2).
This is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above to the cardinal (2(vertical bar xi vertical bar)), the simplest example of a non-(2(vertical bar xi vertical bar))-special tree. As a corollary, we obtain a general result for partially ordered sets:
THEOREM. Let kappa be any infinite regular cardinal, let xi be any ordinal such that 2(vertical bar xi vertical bar) < kappa, and let kappa be any natural number. Let P be a partially ordered set such that P -> (2(vertical bar xi vertical bar))(2<kappa)(1). Then
P -> (kappa + xi)(k)(2).

• 出版日期2014-12