Martin’s Axiom and the ordinal partition relation

Determine whether Martin’s Axiom for aleph_1 together with 2^{aleph_0}=aleph_2 implies the positive ordinal partition relation omega_1^2longrightarrow(omega_1^2,3)^2.

Background

The paper studies the Erdős–Hajnal problem concerning whether the positive partition relation omega_12longrightarrow(omega_12,3)2 follows from Martin’s Axiom at aleph_1 and the continuum hypothesis failure 2{aleph_0}=aleph_2. Hajnal proved that CH implies the corresponding negative relation, while the paper establishes only a relative-consistency result for a restricted form of Martin’s Axiom, namely MA_{aleph_1}(sigma-centered). Thus, the full implication stated above remains unresolved.

References

Erdős and Hajnal have asked whether MA_{\aleph_1}+2{\aleph_0}=\aleph_2 implies eq:main-positive; see . The question is still open.

eq:main-positive:

2⟶(2,3)2.^2\longrightarrow(^2,3)^2.

— Martin's axiom and $ω_1^2 \longrightarrow (ω_1^2, 3)^2$  (2608.13213 - Golshani, 13 Aug 2026) in Section 1, Introduction