Partition theorems for Ketonen-Solovay largeness
Quentin Le Houérou, Ludovic Patey
TL;DR
The paper develops a comprehensive framework for Ketonen–Solovay α-largeness with α<ε_0 and derives sharp partition and Ramsey-type results. By building a Hardy-like hierarchy tied to fundamental sequences, it links α-large sets to explicit growth measures and proves a partition theorem that combines β-large and γ-large pieces via the Hessenberg (natural) sum. It then advances the analysis to α<ω^ω, introducing construction, deconstruction, and sparsity techniques to achieve tight bounds for Ramsey-type largeness, notably proving that ω^{kn+3}-large sets with suitable minimums are RT^2_k-ω^n-large, with the k=2 case giving ω^{2n+3}-largeness and improvements over prior bounds. The work integrates Erdős–Moser and transitive Ramsey principles through a decomposition/grouping strategy, yielding explicit, near-optimal bounds and clarifying the proof-theoretic strength of α-largeness in combinatorial settings. This provides precise conservation-like bounds and strengthens the connections between α-largeness, fast-growing hierarchies, and Ramsey theory.
Abstract
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.