Some logically weak Ramseyan theorems

TL;DR

This paper investigates the logical strength of several Ramseyan theorems within reverse mathematics, demonstrating that most are weaker than ACA_0 and establishing their strong cone avoidance properties.

Contribution

It introduces the Achromatic Ramsey Theorem and proves most related theorems are strictly weaker than ACA_0, revealing their close relationships and cone avoidance properties.

Findings

Most theorems are weaker than ACA_0 over RCA_0

Achromatic Ramsey Theorem has the strong cone avoidance property

Free Set, Thin Set, and Rainbow Ramsey Theorems also have the strong cone avoidance property

Abstract

We study four families of consequences of Ramsey's Theorem from the viewpoint of reverse mathematics. The first, which we call the Achromatic Ramsey Theorem, is from a partition relation introduced by Erd\H{o}s, Hajnal and Rado: $\omega \to [\omega]^r_{c,\leq d}$, which asserts that for every $f: [\omega]^r \to c$ there exists an infinite $H$ with $|f([H]^r)| \leq d$. The second and third are the Free Set Theorem and the Thin Set Theorem, which were introduced by Harvey Friedman. And the last is the Rainbow Ramsey Theorem. We show that, most theorems from these families are quite weak, i.e., they are strictly weaker than $\operatorname{ACA}_0$ over $\operatorname{RCA}_0$. Interestingly, these families turn out to be closely related. We establish the so-called strong cone avoidance property of most instances of the Achromatic Ramsey Theorem by an induction of exponents, then apply this and a similar induction to obtain the strong cone avoidance property of the Free Set Theorem. From the strong cone avoidance property of the Achromatic Ramsey Theorem and the Free Set Theorem, we derive the strong cone property of the Thin Set Theorem and the Rainbow Ramsey Theorem. It follws easily that a theorem with the strong cone avoidance property does not imply $\operatorname{ACA}_0$ over $\operatorname{RCA}_0$.