Consistency strength of negative partition relations

Determine the consistency strength of the negative partition relation \(\lambda^+\nrightarrow(\lambda^+,(3)_{\lambda})^2\) together with \(2^\lambda>\lambda^+\), both for arbitrary strong limit singular \(\lambda\), for \(\lambda=\aleph_\omega\), and simultaneously at every strong limit singular cardinal.

Background

The constructions in the paper begin from substantially strong large-cardinal hypotheses, including supercompact cardinals, while the failure of SCH used in the arguments is known to require weaker but still significant hypotheses such as a measurable cardinal of Mitchell order κ++\kappa^{++}.

This leaves open the exact consistency strength of the resulting negative arrow relations, including the specialized ℵω\aleph_\omega case and the global assertion that the relation holds at every strong limit singular cardinal while SCH fails everywhere at those cardinals.

References

We conclude with a couple of open problems. Maybe the most interesting problem which issues from our study is whether the negative relation holds in \textsf{ZFC}. We believe that the positive relation $\lambda+\rightarrow(\lambda+,(3)_{(\lambda)})2$ is consistent, but we do not know how to prove this: Is it consistent that $\lambda$ is a strong limit singular cardinal and $\lambda+\rightarrow(\lambda+,(3)_{(\lambda)})2$? Is it forceable at $\lambda=\aleph_\omega$?

— On a problem of Erdos and Hajnal  (2502.16625 - Garti et al., 23 Feb 2025) in Question 2.19 (labelled qstrength), end of Section 2