Rainbow Ramsey Theorem for triples is strictly weaker than the Arithmetical Comprehension Axiom

Abstract

We prove that $\operatorname{RCA}_0 + \operatorname{RRT}^3_2 \nvdash \operatorname{ACA}_0$ where $\operatorname{RRT}^3_2$ 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 $\operatorname{RCA}_0 + \operatorname{RRT}^4_2 \vdash \operatorname{ACA}_0$ and obtain some partial answer.