Well-quasi-orders on finite trees and transfinite sequences

Abstract

We study the well-quasi-order (wqo) consisting of the set of finite trees with leaf labels coming from an arbitrary wqo $Q$, ordered by tree homomorphisms which respect the order on the labels. This is a variant of the usual Kruskal tree ordering without infima preservation. We calculate the precise maximal order types of this class of wqos as a function of the maximal order type of the labels $Q$. In the process, we sharpen some recent results of Friedman and Weiermann. Furthermore, we show a correspondence with indecomposable transfinite sequences with finite range, over elements of the wqo $Q$, of length less than $\omega^\omega$. Nash-Williams proved that arbitrary transfinite sequences with finite range are also well-quasi-ordered, but there are no known methods to extract bounds on the maximal order type from the proof. More concrete proofs for sequences of length less than $\alpha$ for some $\alpha < \omega^\omega$ were given by Erd\H{o}s and Rado. Using the correspondence, we obtain precise bounds for the entire collection of transfinite sequences with finite range of length less than $\omega^\omega$.