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 $\mathsf{RT}^{!\alpha}_k$ 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 $k \geq 2$, 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.