  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.
