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

We prove that $\text{RCA}+{\text{RRT}}_2^3⊬\text{ACA}$ where ${\text{RRT}}_2^3$ 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 $\text{RCA}+{\text{RRT}}_2^4⊢\text{ACA}$ and obtain some partial answer.