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

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

Department dc:contributor.department

Mathematics

Year dc:date.issued

2014

Author dc:creator

• Brodsky, Meir Ari

Advisor dc:contributor.advisor

• Todorcevic, B Stevo

• elementary submodels
• nonreflecting ideals
• nonspecial trees
• partial orders
• partition rrelations
• stationary subtrees

Handle dc:identifier.uri

http://hdl.handle.net/1807/68124

OAI identifier oai:identifier

oai:utoronto.scholaris.ca:1807/68124