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 $\kappa$ be any infinite regular cardinal, let $\xi$ be any ordinal such that $2^{|\xi |}<\kappa$, and let $k$ be any natural number. Then \[ \text{non-$\left(2^{<\kappa }\right)$-special tree } \to \left(κ+ ξ\right)^2_k. \] This is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above to the cardinal $(2^{<\kappa }{)}^{+}$, the simplest example of a non-$(2^{<\kappa })$-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 $2^{|\xi |}<\kappa$, and let $k$ be any natural number. Let $P$ be a partially ordered set such that $P\to (2^{<\kappa }{)}_{2^{<\kappa }}^1$. Then \[ P \to \left(κ+ ξ\right)^2_k. \]

arXiv commentSubmitted to Acta Mathematica Hungarica

References in corpus (1)

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

Cited by in corpus (1)

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