Higman's lemma is stronger for better quasi orders

TL;DR

This paper demonstrates that Higman's lemma has greater strength when applied to better quasi orders compared to well quasi orders, establishing a connection with the infinite Ramsey theorem within reverse mathematics.

Contribution

It shows that Higman's lemma is strictly stronger for better quasi orders and links it to the infinite Ramsey theorem in the framework of reverse mathematics.

Findings

Higman's lemma is stronger for better quasi orders.

The infinite Ramsey theorem follows from a property of arrays in well orders.

The results are established within the base theory RCA_0.

Abstract

We prove that Higman's lemma is strictly stronger for better quasi orders than for well quasi orders, within the framework of reverse mathematics. In fact, we show a stronger result: the infinite Ramsey theorem (for tuples of all lengths) follows from the statement that any array $[\mathbb N]^{n+1}\to\mathbb N^n\times X$ for a well order $X$ and $n\in\mathbb N$ is good, over the base theory $\mathsf{RCA_0}$.