We develop the framework of $α$-largeness introduced by Ketonen and Solovay, by proving a partition theorem for $α$-large sets with $α< ε_0$ which generalizes theorems from Ketonen and Solovay and from Bigorajska and Kotlarski. We also prove that for every $ω^{nk+3}$-large set $X$ with $\min X \geq 18$, every coloring $f : [X]^2 \to k$ admits an $ω^n$-large $f$-homogeneous subset. This bound is tight, up to an additive constant.