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

Abstract

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 $κ$ be any infinite regular cardinal, let $ξ$ be any ordinal such that $2^{\left|ξ\right|} < κ$, and let $k$ be any natural number. Then \[ \text{non-$\left(2^{<κ}\right)$-special tree } \to \left(κ+ ξ\right)^2_k. \] 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 $2^{\left|ξ\right|} < κ$, and let $k$ be any natural number. Let $P$ be a partially ordered set such that $P \to (2^{<κ})^1_{2^{<κ}}$. Then \[ P \to \left(κ+ ξ\right)^2_k. \]