Abstract We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a definable set. We prove that some consequences of Ramsey’s Theorem for colorings of pairs could help in simplifying the definitions of some ${\rm{\Delta }}_2^0$ sets, while some others could not. We also investigate some consequences of Ramsey’s Theorem for colorings of longer tuples. These results of definability strength have some interesting consequences in reverse mathematics, including strengthening of known theorems in a more uniform way and also new theorems.

Abstract We prove that RCA0 + RRT ⊬ ACA0 where RRT is the Rainbow Ramsey Theorem for 2-bounded colorings of triples. This reverse mathematical result is based on a cone avoidance theorem, that every 2-bounded coloring of pairs admits a cone-avoiding infinite rainbow, regardless of the complexity of the given coloring. We also apply the proof of the cone avoidance theorem to the question whether RCA0 + RRT ⊦ ACA0 and obtain some partial answer.

Abstract A set of natural numbers B is computably enumerable in and strictly above (or c.e.a. for short) another set C if C