Consistency of tiltan with failure of SCH

Determine whether the tiltan principle \(\clubsuit(\lambda)\) is consistent with \(2^\lambda>\lambda^+\) when \(\lambda\) is a strong limit singular cardinal.

Background

The paper proves that the relevant tiltan principle ♣(λ)\clubsuit(\lambda) yields the negative relation λ+↛(λ+,(3)λ)2\lambda^+\nrightarrow(\lambda^+,(3)_{\lambda})^2. If tiltan could coexist with failure of SCH at a strong limit singular cardinal, it would provide an alternative route to the consistency results developed through pcf methods.

The authors identify the size λ\lambda of the guessing sets in ♣(λ)\clubsuit(\lambda) as the central difficulty.

References

However, we do not know whether \clubsuit(\lambda) is consistent with 2\lambda>\lambda+ when \lambda is a strong limit singular cardinal.

— On a problem of Erdos and Hajnal  (2502.16625 - Garti et al., 23 Feb 2025) in Section 0, Introduction, paragraph beginning “There is another possible approach”