Reductions of well-ordering principles to combinatorial theorems

Abstract

A well-ordering principle is a principle of the form: If $X$ is well-ordered then $F(X)$ is well-ordered, where $F$ is some natural operator transforming linear orders into linear orders. Many important subsystems of Second-order Arithmetic of interest in Reverse Mathematics are known to be equivalent to well-ordering principles. We give a unified treatment for proving lower bounds on the logical strength of various Ramsey-theoretic principles relations using characterizations of the corresponding formal systems in terms of well-ordering principles. Our implications (over $RC{A}_0$) from combinatorial theorems to $AC{A}_0$ and $AC{A}_0^{+}$ also establish uniform computable reductions of the corresponding well-ordering principles to the corresponding Ramsey-type theorems.