1 research outputs found
A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees
Building on early work by Stevo Todorcevic, we describe 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 be any infinite regular cardinal, let be any
ordinal such that , and let be any natural
number. Then
This is a generalization to trees of the Balanced
Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above
to the cardinal , the simplest example of a
non--special tree.
As a corollary, we obtain a general result for partially ordered sets:
Theorem: Let be any infinite regular cardinal, let be any
ordinal such that , and let be any natural
number. Let be a partially ordered set such that . Then Comment: Submitted to Acta Mathematica Hungaric