The strength of Ramsey Theorem for coloring relatively large sets

TL;DR

The paper analyzes RT(!\omega), a Ramsey-type principle for coloring exactly large sets, and proves it is equivalent (over $RCA_0$) to closure under the $\omega$-th Turing jump, i.e., $\forall X\exists Y\,(Y= X^{(\omega)})$. It exhibits computable instances whose homogeneous sets realize $0^{(\omega)}$ and establishes tight lower and upper bounds, connecting the result to both computability theory and proof theory (via ACA$_0^+$ and truth-predicate frameworks). It also shows a regressive variant KM(!\omega) is equivalent to RT(!\omega) and links the principle to Peano Arithmetic with an $\omega$-indexed hierarchy of truth predicates, identifying the arithmetical strength with transfinite induction up to $\varphi_2(0)$. The work suggests broad transfinite generalizations to $\alpha$-large sets and $\alpha$-jumps, with potential implications for ATR$_0$ and finite independence phenomena related to Paris–Harrington style results.

Abstract

We characterize the computational content and the proof-theoretic strength of a Ramsey-type theorem for bi-colorings of so-called {\em exactly large} sets. An {\it exactly large} set is a set $X\subset\Nat$ such that $\card(X)=\min(X)+1$. The theorem we analyze is as follows. For every infinite subset $M$ of $\Nat$, for every coloring $C$ of the exactly large subsets of $M$ in two colors, there exists and infinite subset $L$ of $M$ such that $C$ is constant on all exactly large subsets of $L$. This theorem is essentially due to Pudlàk and Rödl and independently to Farmaki. We prove that --- over Computable Mathematics --- this theorem is equivalent to closure under the $ω$ Turing jump (i.e., under arithmetical truth). Natural combinatorial theorems at this level of complexity are rare. Our results give a complete characterization of the theorem from the point of view of Computable Mathematics and of the Proof Theory of Arithmetic. This nicely extends the current knowledge about the strength of Ramsey Theorem. We also show that analogous results hold for a related principle based on the Regressive Ramsey Theorem. In addition we give a further characterization in terms of truth predicates over Peano Arithmetic. We conjecture that analogous results hold for larger ordinals.

Theorems & Definitions (35)

• Theorem 1: Pudlàk-Rödl Pud-Rod:82 and Farmaki Far:taFar-Neg:08
• proof : Proof of Theorem \ref{['thm:rtomega']}
• Theorem 2: Jockusch, Joc:72
• Theorem 3: Simpson, Sim
• Theorem 4: McAloon, McA:85
• Proposition 1
• proof
• Theorem 5
• Lemma 1
• proof