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

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