Degrees bounding principles and universal instances in reverse mathematics

Abstract

A Turing degree d bounds a principle P of reverse mathematics if every computable instance of P has a d-computable solution. P admits a universal instance if there exists a computable instance such that every solution bounds P. We prove that the stable version of the ascending descending sequence principle (SADS) as well as the stable version of the thin set theorem for pairs (STS(2)) do not admit a bound of low_2 degree. Therefore no principle between Ramsey's theorem for pairs RT22 and SADS or STS(2) admit a universal instance. We construct a low_2 degree bounding the Erd\H{o}s-Moser theorem (EM), thereby showing that previous argument does not hold for EM. Finally, we prove that the only Delta^0_2 degree bounding a stable version of the rainbow Ramsey theorem for pairs (SRRT22) is 0'. Hence no principle between the stable Ramsey theorem for pairs SRT22 and SRRT22 admit a universal instance. In particular the stable version of the Erd\H{o}s-Moser theorem does not admit one. It remains unknown whether EM admits a universal instance.