Rainbow Ramsey theorem for triples is strictly weaker than the Arithmetic Comprehension Axiom

TL;DR

The paper demonstrates that the Rainbow Ramsey Theorem for triples does not imply the Arithmetic Comprehension Axiom within reverse mathematics, using a cone avoidance approach to establish the result.

Contribution

It introduces a cone avoidance theorem for 2-bounded colorings, showing the Rainbow Ramsey Theorem for triples is strictly weaker than ACA in reverse mathematics.

Findings

RCA + RRT^3_2 does not prove ACA

Every 2-bounded coloring of pairs admits a cone-avoiding infinite rainbow

Partial results on RRT^4_2 and ACA relationship

Abstract

We prove that $\RCA + \RRT^3_2 \not\vdash \ACA$ where $\RRT^3_2$ is the Rainbow Ramsey Theorem for 2-bounded colorings of triples. This reverse mathematical result is based on a cone avoidance theorem, that every 2-bounded coloring of pairs admits a cone-avoiding infinite rainbow, regardless of the complexity of the given coloring. We also apply the proof of the cone avoidance theorem to the question whether $\RCA + \RRT^4_2 \vdash \ACA$ and obtain some partial answer.