Cardinal characteristics at {\kappa} in a small u(\kappa) model

TL;DR

This paper constructs a model with a supercompact cardinal where the minimal size of a certain ultrafilter base is less than the continuum, and explores related cardinal characteristics.

Contribution

It provides a complete proof of a model where u(κ) < 2^κ for supercompact κ, using a novel modification of existing constructions and analyzing related cardinal invariants.

Findings

Established a model with u(κ) < 2^κ for supercompact κ

Generalized classical facts and forcing notions to uncountable cardinals

Studied the behavior of generalized cardinal characteristics in the model

Abstract

We provide a model where u(\kappa) < 2^{\kappa} for a supercompact cardinal \kappa. Garti and Shelah have provided a sketch of how to obtain such a model by modifying the construction in a paper of Dzamonja and Shelah; we provide here a complete proof using a different modification and further study the values of other natural generalizations of classical cardinal characteristics in our model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.