Partition theorems for Ketonen-Solovay largeness

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.

💡 Research Summary

This paper develops a comprehensive theory of α‑largeness, a combinatorial notion introduced by Ketonen and Solovay to measure the size of finite sets in weak arithmetic. The authors first formalize α‑largeness for every ordinal α < ε₀ using fundamental sequences {α}(n). A finite ordered set X={x₀<…<x_{s}} is α‑large if the iterated application {α}(X) = …{{α}(x₀)}(x₁)…(x_{s}) yields 0; otherwise it is α‑small. “Exactly α‑large” means α‑large and the set without its maximum is α‑small, while “at most α‑large” requires only the latter condition.

The central combinatorial contribution is Main Theorem 1.1: if B and C are respectively at most β‑large and at most γ‑large for ordinals β, γ < ε₀, then their union B∪C is at most (β ⊕ γ)‑large, where ⊕ denotes the natural (Hessenberg) sum of ordinals. This result unifies and extends several earlier statements: Ketonen‑Solovay’s Lemma 4.6, the pigeon‑hole theorems of Bigorajska‑Kotlarski for the “star” version of largeness, and various structural lemmas concerning fundamental sequences. The proof relies on a careful analysis of pseudo‑norms psn(α), which capture the maximal coefficient or exponent appearing in the Cantor normal form of α, and on a “Hardy‑like” hierarchy of fast‑growing functions h_α. The authors introduce a relation β ⇒_x α, which refines ordinary ordinal comparison when the argument x is small (≤ psn(α)). Lemmas 2.12–2.14 show that this relation guarantees monotonicity and domain inclusion for the h_α functions, allowing one to compare h_β(x) and h_α(x) even when β ≫ α but x is not large enough for the simple inequality β ≥ α to hold.

The second major achievement is Main Theorem 1.2, a tight Ramsey‑type closure result for α‑largeness when α < ω^ω. For any integers n ≥ 1, k ≥ 1, and any finite set X⊆ℕ that is ω^{nk+3}‑large with min X ≥ 18, every k‑coloring f: