Somewhere over the rainbow Ramsey theorem for pairs

Abstract

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.