Stick, a Gal ultrafilter, and increased power-set size

Determine whether there exists a model in which λ is supercompact, a normal ultrafilter 𝒰 over λ satisfies Gal(𝒰,λ^+,λ^+), the stick principle (λ) holds, and 2^λ>λ^+.

Background

The final section proposes combining a normal ultrafilter with the Gal property and Prikry-type forcing. The Gal property is intended to ensure that large sets in the generic extension contain large ground-model subsets, which can preserve stick or tiltan sequences through the forcing.

The paper notes that the ingredients can be obtained by several separate methods, but the simultaneous realization of the Gal ultrafilter property, stick at λ^+, and failure of the local GCH at λ is not known. The question therefore targets a combined consistency construction at a supercompact cardinal.

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 (label qstick), end of Section 3

Stick, a Gal ultrafilter, and increased power-set size

Background

References

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