Bounding finite-image sequences of length $ω^k$

Given a well-quasi-order $X$ and an ordinal $α$, the set $s^F_α(X)$ of transfinite sequences on $X$ with length less than $α$ and with finite image is also a well-quasi-order, as proven by Nash-Williams. Before Nash-Williams proved it for general $α$, however, it was proven for $α<ω^ω$ by Erdős and Rado. In this paper, we revisit Erdős and Rado’s proof and improve upon it, using it to obtain upper bounds on the maximum linearization of $s^F_{ω^k}(X)$ in terms of $k$ and $o(X)$, where $o(X)$ denotes the maximum linearization of $X$. We show that, for fixed $k$, $o(s^F_{ω^k}(X))$ is bounded above by a function which can roughly be described as $(k+1)$-times exponential in $o(X)$. We also show that, for $k\le 2$, this bound is not far from tight.

💡 Research Summary

The paper investigates the maximal linearization (type) o(sᶠ_α(X)) of the well‑quasi‑ordered set of finite‑image transfinite sequences sᶠ_α(X) over a well‑quasi‑order X, focusing on the case α = ωᵏ for a fixed natural number k. While Nash‑Williams (1965) proved that sᶠ_α(X) is a WQO for any ordinal α, he did not give quantitative bounds on its type in terms of α and o(X). Earlier, Erdős and Rado handled the special case α < ω^ω, but their proof did not yield explicit bounds.

The authors revisit the Erdős‑Rado argument and refine it to obtain a concrete upper bound that is substantially tighter than the naïve bound one would get by iterating the operation X ↦ X* k times (which leads to a tower of 2k exponentials). Their improvement rests on two key modifications:

1. Re‑ordering of the decomposition – Instead of viewing an ωᵏ‑length sequence as a sequence of ω‑blocks over an alphabet of ω^{k‑1}‑length sequences (the original Erdős‑Rado approach), they treat it as a sequence of ω‑blocks over an alphabet consisting of ω^{k‑1}‑length sequences. This reversal simplifies the inductive construction.

2. Use of finite‑power set construction only once – The original proof repeatedly applied the “star” operation (X ↦ X*) at each level, producing a tower of 2k exponentials. The new proof applies the star operation only at the base level and then relies on the finite‑power‑set operator ℘′_fin (the non‑empty finite subsets) for the remaining levels. Since ℘′_fin has a known bound o(℘_fin(Y)) ≤ 2·o(Y), this reduces the height of the exponential tower by one.

To formalize the construction, the authors define auxiliary WQOs P₀, P₁,…, P_k and Q₁,…, Q_k:

• P₀(X) = X.
• Q_k = ⊔_{i=0}^{k‑1} P_i (disjoint union).
• P_k = ℘′_fin(Q_k) (non‑empty finite subsets of Q_k).

Using a family of monotone maps φ_X they embed each P_k into sᶠ_{ω^{k+1}}(X) and each Q_k into sᶠ_{ω^{k‑1}+1}(X). The map φ_X for P_k is defined by taking a finite set S = { s₀,…,s_{p‑1} }⊂Q_k and sending it to the ω‑power of the concatenation of the images of its elements: \