Partition theorems for Ketonen-Solovay largeness

Abstract

We develop the framework of $\alpha$-largeness introduced by Ketonen and Solovay, by proving a partition theorem for $\alpha$-large sets with $\alpha < \epsilon_0$ which generalizes theorems from Ketonen and Solovay and from Bigorajska and Kotlarski. We also prove that for every $\omega^{nk+3}$-large set $X$ with $\min X \geq 18$, every coloring $f : [X]^2 \to k$ admits an $\omega^n$-large $f$-homogeneous subset. This bound is tight, up to an additive constant.