Positive partition relation at strong-limit singular cardinals

Determine whether it is consistent that a strong limit singular cardinal \(\lambda\) satisfies \(\lambda^+\rightarrow(\lambda^+,(3)_{\lambda})^2\), and determine whether such a positive relation is forceable when \(\lambda=\aleph_\omega\).

Background

The paper proves consistency of the corresponding negative relation with failure of the singular cardinal hypothesis, namely λ+↛(λ+,(3)λ)2\lambda^+\nrightarrow(\lambda^+,(3)_{\lambda})^2 together with 2λ>λ+2^\lambda>\lambda^+. It then asks whether the opposite, positive relation can consistently hold under the same broad cardinal-arithmetic setting.

The authors note that any positive consistency result would need to add many bounded subsets to λ\lambda, eliminating GCH along every unbounded sequence of cardinals below λ\lambda whose true cofinality is λ+\lambda^+.

References

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 1.4 (labelled qeh5positive), end of Section 1

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 1.1 (labelled qeh5positive), Section 1, conclusion of Section 1