  We study the strength of $\RRT^3_2$, Rainbow Ramsey Theorem for colorings of
triples, and prove that $\RCA + \RRT^3_2$ implies neither $\WKL$ nor
$\RRT^4_2$. To this end, we establish some recursion theoretic properties of
cohesive sets and rainbows for colorings of pairs. We show that every sequence
(2-bounded coloring of pairs) admits a cohesive set (infinite rainbow) of
non-PA Turing degree; and that every $\emptyset'$-recursive sequence (2-bounded
coloring of pairs) admits a $\low_3$ cohesive set (infinite rainbow).
