Bounded Ramsey's theorem for triples in computability theory

TL;DR

This paper investigates a bounded version of Ramsey's theorem for triples, showing it shares computational bounds with the classic pairs version but is not reducible to it, revealing nuanced differences in their computational content.

Contribution

It introduces a bounded restriction of Ramsey's theorem for triples and demonstrates its computational properties are similar yet distinct from the classical pairs version.

Findings

Shares computability-theoretic upper bounds with RT^2_2

Not computably reducible to RT^2_2 even with multiple applications

Highlights nuanced differences in computational content between bounded triples and pairs

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 $\mathsf{RT}^2_2$.