Finitary Ramsey implications

Determine whether I-Sigma-sub-n-plus-one together with WPHP at level Sigma-n-plus-one proves the stated finite Ramsey theorem FRT-two-two, and whether FRT-two-two proves FRT-two-three.

Background

The paper records known separations involving WPHP, generalized pigeonhole principles, and finite Ramsey principles. It then asks for two unresolved implications between the specified Ramsey-theoretic statements.

References

$\text{I}\Sigma_{n+1} + \text{WPHP}(\Sigma_{n+1}) \vdash \text{FRT}22(\Sigma{n+1})$? $\text{FRT}22(\Sigma{n+1}) \vdash \text{FRT}23(\Sigma{n+1})$?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in 2025 Section 5