ON THE ROLE OF THE COLLECTION PRINCIPLE FOR Σ2-FORMULAS IN SECOND-ORDER REVERSE MATHEMATICS

We show that the principle PART from Hirschfeldt and Shore is equivalent to the Σ2-Bounding principle BΣ 0 2 over RCA0, answering one of their open questions. Furthermore, we also fill a gap in a proof of Cholak, Jockusch and Slaman by showing that D2 2 implies BΣ 0 2 and is thus indeed equivalent to Stable Ramsey’s Theorem for Pairs (SRT2). This also allows us to conclude that the combinatorial principles IPT2, SPT 2 2 and SIPT 2 2 defined by Dzhafarov and Hirst all imply BΣ2 and thus that SPT 2 2 and SIPT 2 2 are both equivalent to SRT 2 2 as well. Our proof uses the notion of a bi-tame cut, the existence of which we show to be equivalent, over RCA0, to the failure of BΣ2.