Partition theorems for Ketonen-Solovay largeness

TL;DR

This paper extends the theory of $eta$-largeness by proving a partition theorem for $eta$-large sets with $eta < oldsymbol{ ext{ extepsilon}}_0$, and establishes tight bounds for homogeneous subsets under colorings.

Contribution

It generalizes existing theorems on $eta$-largeness and homogeneous sets, providing new partition results and bounds for $eta$-large sets with $eta < oldsymbol{ ext{ extepsilon}}_0$.

Findings

Proved a partition theorem for $eta$-large sets with $eta < oldsymbol{ ext{ extepsilon}}_0$.

Established tight bounds for homogeneous subsets in colored $eta$-large sets.

Extended the framework of $eta$-largeness introduced by Ketonen and Solovay.

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.