On the role of the collection principle for Σ⁰₂-formulas in second-order reverse mathematics

We show that the principle  P A R T \mathsf {PART} from Hirschfeldt and Shore is equivalent to the Σ 2 0 \Sigma ^0_2 -Bounding principle  B Σ 2 0 B\Sigma ^0_2 over  R C A 0 \mathsf {RCA}_0 , answering one of their open questions. Furthermore, we also fill a gap in a proof of Cholak, Jockusch and Slaman by showing that D 2 2 D^2_2 implies  B Σ 2 0 B\Sigma ^0_2 and is thus indeed equivalent to Stable Ramsey’s Theorem for Pairs ( S R T 2 2 \mathsf {SRT}^2_2 ). This also allows us to conclude that the combinatorial principles I P T 2 2 \mathsf {IPT}^2_2 , S P T 2 2 \mathsf {SPT}^2_2 and  S I P T 2 2 \mathsf {SIPT}^2_2 defined by Dzhafarov and Hirst all imply  B Σ 2 0 B\Sigma ^0_2 and thus that S P T 2 2 \mathsf {SPT}^2_2 and  S I P T 2 2 \mathsf {SIPT}^2_2 are both equivalent to  S R T 2 2 \mathsf {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  R C A 0 \mathsf {RCA}_0 , to the failure of  B Σ 2 0 B\Sigma ^0_2 .