Consistency of a small ultrafilter number at ω₁

Determine whether it is consistent (relative to ZFC or suitable large-cardinal assumptions) that the ultrafilter number at ω₁ satisfies the strict inequality u_{ω₁} < 2^{ω₁}.

Background

The generalized ultrafilter number at a cardinal κ, denoted u_κ, is the minimum size of a base generating a uniform ultrafilter on κ. For κ=ω the classical ultrafilter number u has been extensively studied, and techniques involving Mathias forcing are known to separate u from the continuum. However, analogous separations at higher cardinals present significant challenges.

Within the paper’s overview of generalized cardinal characteristics and the Tukey order on ultrafilters, the authors note that the specific case κ=ω₁ remains unresolved: whether there exists a model in which u_{ω₁} is strictly below 2^{ω₁}. This is identified as a long-standing open problem in the literature.