Stick, a Galvin ultrafilter, and failure of SCH

Determine whether it is consistent that a cardinal \(\lambda\) is supercompact, a normal ultrafilter \(\mathscr U\) on \(\lambda\) satisfies \({\rm Gal}(\mathscr U,\lambda^+,\lambda^+)\), the stick principle \(\clubsuit(\lambda)\) holds, and \(2^\lambda>\lambda^+\).

Background

The paper shows that the stick principle ♣(λ)\clubsuit(\lambda) implies the desired negative partition relation. It discusses forcing ♣(λ)\clubsuit(\lambda) while increasing 2λ2^\lambda, and separately describes a Galvin-type property for a normal ultrafilter that can preserve stick or tiltan through Prikry-type forcing.

The unresolved issue is whether these ingredients can be forced together: supercompactness, the Galvin ultrafilter property, stick at λ+\lambda^+, and failure of SCH at λ\lambda.

References

There are also several ways to force stick or tiltan at a supercompact cardinal \lambda while increasing 2\lambda above \lambda+. We do not know, however, to force these two things together: Is it consistent that \lambda is supercompact, \mathscr{U} is a normal ultrafilter over \lambda satisfying {\rm Gal}(\mathscr{U},\lambda+,\lambda+), (\lambda) holds and 2\lambda>\lambda+?

— On a problem of Erdos and Hajnal  (2502.16625 - Garti et al., 23 Feb 2025) in Question 3.1 (labelled qstick), Section 3, final paragraph

Though we do not know how to force the existence of wondrous ideals, we can prove (from the results of the previous section) that in some sense there are no such ideals over two consecutive cardinals.

— Quadruples and cubes  (2609.11239 - Garti, 10 Sep 2026) in Section 2, paragraph immediately preceding Theorem 2.11