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
.