Consistency strength of negative partition relations with failure of SCH

Determine the consistency strength of the negative partition relation λ^+↛(λ^+,(3)_{(λ)})^2 together with 2^λ>λ^+ for (i) an arbitrary strong limit singular cardinal λ, (ii) λ=aleph_ω, and (iii) every strong limit singular cardinal λ simultaneously in a universe satisfying 2^λ>λ^+ at every such cardinal.

Background

The constructions in the paper obtain the negative relation with 2λ>λ+ from substantial large-cardinal assumptions, beginning with a supercompact cardinal in the ground model. The authors note that the failure of SCH at λ requires, by a result of Gitik, a measurable cardinal κ with o(κ)=κ{++}. This leaves a gap between the large-cardinal strength used in the construction and the lower strength known to be necessary for the relevant cardinal-arithmetic behavior.

The question asks for the exact or optimal consistency strength in three increasingly demanding settings: one strong limit singular cardinal, the specific case aleph_ω, and all strong limit singular cardinals simultaneously.

References

The gap between these large cardinals invites the following: Let $\lambda$ be a strong limit singular cardinal. What is the consistency strength of the negative arrow relation $\lambda+\nrightarrow(\lambda+,(3)_{(\lambda)})2$ with $2\lambda>\lambda+$? What is the consistency strength of the same negative relation with $2\lambda>\lambda+$ where $\lambda=\aleph_\omega$? What is the consistency strength of the negative relation at every strong limit singular cardinal $\lambda$, in a universe in which $2\lambda>\lambda+$ at every such $\lambda$?

— On a problem of Erdos and Hajnal  (2502.16625 - Garti et al., 23 Feb 2025) in Question 2.1 (label qstrength), end of Section 2