Cohesive sets and rainbows

We study the strength of RRT23, Rainbow Ramsey Theorem for colorings of triples, and prove that RCA0+RRT23 implies neither WKL0 nor RRT24. 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 ∅′-recursive sequence (2-bounded coloring of pairs) admits a low3 cohesive set (infinite rainbow).