The strength of Ramsey's theorem for $\alpha$-large sets

Abstract

We calibrate the reverse mathematical strength of a family of extensions of Ramsey's theorem to finite colorings of certain subsets of the natural numbers of unbounded finite dimension. Specifically, we analyze the principles ${\mathsf{RT}}_k^{!\alpha }$ asserting that every $k$-coloring of the exactly $\alpha$-large subsets of an infinite $X\subseteq \mathbb{N}$ admits an infinite homogeneous set, where $\alpha$-largeness is defined via systems of fundamental sequences in the style of Ketonen and Solovay. For each countable ordinal $\alpha <{\Gamma }_0$ and each $k\ge 2$, we prove over ${\mathsf{RCA}}_0$ that the hierarchy of theorems $\mathsf{RT}^{!\a}_k$ corresponds exactly to the hierarchy of systems axiomatized by closure under transfinite Turing jumps, yielding a fine-grained classification between ${\mathsf{ACA}}_0$ and ${\mathsf{ATR}}_0$. Our results extend previous work on the case $\alpha =\omega$ and provide a uniform correspondence between countable indecomposable ordinals below ${\Gamma }_0$ and natural Ramsey-like theorems.