SRT22 does not imply RT22 in omega-models

Abstract

We complete a 40-year old program on the computability-theoretic analysis of Ramsey's theorem, starting with Jockusch in 1972, and improving a result of Chong, Slaman and Yang in 2014. Given a set $X$, let $[X]^n$ be the collection of all $n$-element subsets of $X$. Ramsey's theorem for $n$-tuples asserts the existence, for every finite coloring of $[\omega]^n$, of an infinite set $X \subseteq \omega$ such that $[X]^n$ is monochromatic. The meta-mathematical study of Ramsey has a rich history, with several long-standing open problems and seminal theorems, including Seetapun's theorem in 1995 and Liu's theorem in 2012 about Ramsey's theorem for pairs. The remaining question about the study of Ramsey's theorem from a computational viewpoint was the relation between Ramsey's theorem for pairs ($\mathsf{RT}^2_2$) and its restriction to stable colorings ($\mathsf{SRT}^2_2$), that is, colorings admitting a limit behavior. Chong, Slaman and Yang first proved that $\mathsf{SRT}^2_2$ does not formally imply $\mathsf{RT}^2_2$ in a proof-theoretic sense, using non-standard models of reverse mathematics. In this article, we answer the open question whether this non-implication also holds within the framework of computability theory. More precisely, we construct a $\omega$-model of $\mathsf{SRT}^2_2$ which is not a model of $\mathsf{RT}^2_2$. For this, we design a new notion of effective forcing refining Mathias forcing using the notion of largeness classes.