  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.
