Higman’s Lemma is Stronger for Better Quasi Orders

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) is equivalent to the statement that any array \([\mathbb N]^{n+1}\rightarrow \mathbb N^n\times X\) for a well order X and \(n\in \mathbb N\) is good, over the base theory \(\mathsf {RCA_0}\).