On the role of the collection principle for $\Sigma ^0_2$-formulas in second-order reverse mathematics

Abstract:

We show that the principle $\mathsf {PART}$ from Hirschfeldt and Shore is equivalent to the $\Sigma ^0_2$-Bounding principle $B\Sigma ^0_2$ over $\mathsf {RCA}_0$, answering one of their open questions.

Furthermore, we also fill a gap in a proof of Cholak, Jockusch and Slaman by showing that $D^2_2$ implies $B\Sigma ^0_2$ and is thus indeed equivalent to Stable Ramsey’s Theorem for Pairs ($\mathsf {SRT}^2_2$). This also allows us to conclude that the combinatorial principles $\mathsf {IPT}^2_2$, $\mathsf {SPT}^2_2$ and $\mathsf {SIPT}^2_2$ defined by Dzhafarov and Hirst all imply $B\Sigma ^0_2$ and thus that $\mathsf {SPT}^2_2$ and $\mathsf {SIPT}^2_2$ are both equivalent to $\mathsf {SRT}^2_2$ as well.

Our proof uses the notion of a bi-tame cut, the existence of which we show to be equivalent, over $\mathsf {RCA}_0$, to the failure of $B\Sigma ^0_2$.