  The rainbow Ramsey theorem states that every coloring of tuples where each
color is used a bounded number of times has an infinite subdomain on which no
color appears twice. The restriction of the statement to colorings over pairs
(RRT22) admits several characterizations: it is equivalent to finding an
infinite subset of a 2-random, to diagonalizing against Turing machines with
the halting set as oracle... In this paper we study principles that are closely
related to the rainbow Ramsey theorem, the Erd\H{o}s Moser theorem and the thin
set theorem within the framework of reverse mathematics. We prove that the thin
set theorem for pairs implies RRT22, and that the stable thin set theorem for
pairs implies the atomic model theorem over RCA. We define different notions of
stability for the rainbow Ramsey theorem and establish characterizations in
terms of Ramsey-type K\"onig's lemma, relativized Schnorr randomness or
diagonalization of Delta2 functions.
