A Ramsey theorem for trees

Abstract

We prove a Ramsey theorem for trees. The infinite version of this theorem can be stated: if T is a rooted tree of infinite height with each node of T having at least one but finitely many immediate successors, if n is a positive integer, and if the collection of all strongly embedded, height-n subtrees of T is partitioned into finitely many classes, then there must exist a strongly embedded subtree S of T with S having infinite height and with all the strongly embedded, height-n subtrees of S in the same class. © 1979.