Abstract. We show that, over the base theory RCA 0 , Stable Ramsey's Theorem for Pairs implies neither Ramsey's Theorem for Pairs nor Σ 0 2 -induction.

No abstract

Our proof uses the notion of a bi-tame cut, the existence of which we show to be equivalent, over RCA 0 , to the failure of BΣ 0 2 .

We study the pigeonhole principle for Σ 2 -definable injections with domain twice as large as the codomain, and the weak König lemma for ∆ 0 2 -definable trees in which every level has at least half of the possible nodes. We show that the latter implies the existence of 2-random reals, and is conservative over the former. We also show that the former is strictly weaker than the usual pigeonhole principle for Σ 2 -definable injections.

We study randomness notions given by higher recursion theory, establishing the relationships Π 1 1 -randomness ⊂ Π 1 1 -Martin-Löf randomness ⊂ ∆ 1 1 -randomness = ∆ 1 1 -Martin-Löf randomness. We characterize the set of reals that are low for ∆ 1 1 randomness as precisely those that are ∆ 1 1 -traceable. We prove that there is a perfect set of such reals.