Consistency strength of negative arrow relations

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

Background

The constructions in the paper begin with a supercompact cardinal, while the failure of SCH used in the forcing arguments requires, by results of Gitik, a measurable cardinal of sufficiently high Mitchell order. The authors point to the gap between these large-cardinal assumptions as motivating a sharper analysis of consistency strength.

The question has three explicitly distinguished parts: the general singular case, the case $\lambda=\aleph_\omega$, and a global assertion covering all strong limit singular cardinals in a universe where SCH fails everywhere at such cardinals.