Consistency of the positive successor-of-singular partition relation

Determine whether it is consistent that a strong limit singular cardinal λ satisfies the positive partition relation λ^+→(λ^+,(3)_{(λ)})^2, and determine whether this positive relation is forceable when λ=aleph_ω.

Background

The paper establishes consistency results for the negative relation λ+↛(λ+,(3)_{(λ)})2 when λ is a strong limit singular cardinal and 2λ>λ+, including a result at λ=aleph_ω. The authors then ask whether the corresponding positive relation can consistently hold. They further isolate the special case of forcing the positive relation at aleph_ω.

The discussion indicates that any positive consistency result would need to add many bounded subsets to λ and eliminate GCH on every unbounded sequence of cardinals below λ whose true cofinality is λ+. Thus the question concerns the compatibility of the positive partition relation with strong-limit singular cardinal arithmetic, rather than merely the existence of a coloring in a fixed model.

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 1.1 (label qeh5positive), end of Section 1