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Bounded Ramsey's theorem for triples in computability theory

Ludovic Patey, Paul Shafer

Abstract

We study a restriction of Ramsey's theorem for 2-coloring of triples, in which homogeneous sets for color~1 are of bounded size ($\mathsf{BRT}^3_2$). We prove that the computational content of this statement is very close to Ramsey's theorem for pairs ($\mathsf{RT}^2_2)$, in that it satisfies the same known computability-theoretic upper bounds, but that $\mathsf{BRT}^3_2$ is not computably-reducible to $\mathsf{RT}^2_2$, even when allowing multiple applications of $\mathsf{RT}^2_2$.

Bounded Ramsey's theorem for triples in computability theory

Abstract

We study a restriction of Ramsey's theorem for 2-coloring of triples, in which homogeneous sets for color~1 are of bounded size (). We prove that the computational content of this statement is very close to Ramsey's theorem for pairs (, in that it satisfies the same known computability-theoretic upper bounds, but that is not computably-reducible to , even when allowing multiple applications of .

Paper Structure

This paper contains 14 sections, 28 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Reverse mathematics and computable reductions
  3. Stability and cohesiveness
  4. Main contributions
  5. $\mathsf{BRT}^3_2$ and the Erdős-Moser theorem
  6. Cone avoidance
  7. Constant-bound trace avoidance
  8. Effective bounds of $\mathsf{BRT}^3_2$
  9. Weakly low basis for $\Delta^0_2$ cohesive $\mathsf{BRT}^2_2$
  10. Weakly low basis for $\Delta^0_2$ stable $\mathsf{BRT}^2_2$
  11. Preserving multiple hyperimmunities
  12. Negative preservations
  13. $\Delta^0_2$ stable $\mathsf{BRT}^2_2$ preserves $\omega$ hyperimmunities
  14. $\mathsf{BRT}^3_{2,4}$ does not computably-reduce to $\mathsf{RT}^2$

Key Result

Lemma 1.2

Let $f : [\mathbb{N}]^{n+1} \to 2$ be a computable stable instance of $\mathsf{BRT}^{n+1}_{2, \ell+1}$. Its limit coloring is a $\Delta^0_2$ instance of $\mathsf{BRT}^n_{2, \ell}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (68)

  • Definition 1.1: Hirschfeldt and Jockusch hirschfeldt_notions_2016
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4: Cholak, Jockusch and Slaman cholak2001strength
  • proof
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • ...and 58 more