Well-quasi-orders on finite trees and transfinite sequences

TL;DR

This paper computes the maximal order type of the wqo on finite leaf-labelled trees $\mathsf{T_f}(Q)$ (ordered by leaf-respecting tree homomorphisms) as a function of the maximal order type $o(Q)$, establishing the sharp bound $o(\mathsf{T_f}(Q)) = g(o(Q))$. The fixed-point-free epsilon-based function $g$ satisfies $g(1)=\omega$ and $g(\alpha) = \bar{\varepsilon}(\alpha-2)$ for $\alpha\ge 2$, and the authors prove both upper and lower bounds that match this value using lower-set decompositions and finite powerset techniques. They further connect leaf-labelled trees to indecomposable transfinite sequences $i^F_{\omega^\omega}(Q)$, showing an exact correspondence with $\mathsf{T_f}(Q)$ and transferring maximal order-type results to transfinite sequences of length $< \omega^\omega$, yielding $o(s^F_{\omega^\omega}(Q)) = g(o(Q))$. This work thus provides precise ordinal bounds for a broad class of wqos and clarifies the structure of transfinite sequence orders, with implications for reverse mathematics and proof-theoretic strength analyses of related combinatorial principles.

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 $ω^ω$. 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 $α$ for some $α< ω^ω$ were given by Erdős and Rado. Using the correspondence, we obtain precise bounds for the entire collection of transfinite sequences with finite range of length less than $ω^ω$.

Figures (3)

• Figure 1: Examples of finite leaf-labelled trees.
• Figure 2: Examples of tree homomorphisms.
• Figure 3: The map $h \colon \omega^2_{\text{desc}} \to \mathsf{T_f}(2)$ for arbitrary $(i,j) \in \omega^2_{\text{desc}}$, and an example of the tree $h((2,1))$.

Theorems & Definitions (109)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Remark 2.7
• Proposition 2.8
• proof
• Definition 2.9