Partition theorems for Ketonen-Solovay largeness

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 $\min X\ge 18$, every coloring $f:[X{]}^2\to k$ admits an ${\omega }^n$-large $f$-homogeneous subset. This bound is tight, up to an additive constant.