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The strength of Ramsey's theorem for $α$-large sets

Lorenzo Carlucci, Andrea Volpi, Konrad Zdanowski

Abstract

We calibrate the reverse mathematical strength of a family of extensions of Ramsey's theorem to finite colorings of certain subsets of the natural numbers of unbounded finite dimension. Specifically, we analyze the principles $\mathsf{RT}^{!α}_k$ asserting that every $k$-coloring of the exactly $α$-large subsets of an infinite $X \subseteq \mathbb{N}$ admits an infinite homogeneous set, where $α$-largeness is defined via systems of fundamental sequences in the style of Ketonen and Solovay. For each countable ordinal $α< Γ_0$ and each $k \geq 2$, we prove over $\mathsf{RCA}_0$ that the hierarchy of theorems $\mathsf{RT}^{!\a}_k$ corresponds exactly to the hierarchy of systems axiomatized by closure under transfinite Turing jumps, yielding a fine-grained classification between $\mathsf{ACA}_0$ and $\mathsf{ATR}_0$. Our results extend previous work on the case $α=ω$ and provide a uniform correspondence between countable indecomposable ordinals below $Γ_0$ and natural Ramsey-like theorems.

The strength of Ramsey's theorem for $α$-large sets

Abstract

We calibrate the reverse mathematical strength of a family of extensions of Ramsey's theorem to finite colorings of certain subsets of the natural numbers of unbounded finite dimension. Specifically, we analyze the principles asserting that every -coloring of the exactly -large subsets of an infinite admits an infinite homogeneous set, where -largeness is defined via systems of fundamental sequences in the style of Ketonen and Solovay. For each countable ordinal and each , we prove over that the hierarchy of theorems corresponds exactly to the hierarchy of systems axiomatized by closure under transfinite Turing jumps, yielding a fine-grained classification between and . Our results extend previous work on the case and provide a uniform correspondence between countable indecomposable ordinals below and natural Ramsey-like theorems.
Paper Structure (12 sections, 32 theorems, 82 equations)

This paper contains 12 sections, 32 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ordinals and fundamental sequences
  4. Coding Turing jumps
  5. Largeness notions based on ordinals
  6. Systems between $\mathsf{ACA}_0$ and $\mathsf{ATR}_0$
  7. Ramsey theory
  8. Lower bound
  9. Coding Turing jumps into homogeneous sets
  10. Proof through well-ordering principles
  11. Upper bound
  12. Conclusions and perspectives

Key Result

Theorem 1.1

For each $\alpha < \Gamma_0$, for all $k\geq 2$,

Theorems & Definitions (79)

  • Theorem 1.1: Main theorem
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Lemma 2.9
  • ...and 69 more