Cohesive sets and rainbows

We study the strength of ${\text{RRT}}_2^3$, Rainbow Ramsey Theorem for colorings of triples, and prove that $\text{RCA}+{\text{RRT}}_2^3$ implies neither $\text{WKL}$ nor ${\text{RRT}}_2^4$. 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 ${\mathrm{\varnothing }}^{\mathrm{\prime }}$-recursive sequence (2-bounded coloring of pairs) admits a ${\text{low}}_3$ cohesive set (infinite rainbow).