  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.
