Cofinal types of ultrafilters on measurable cardinals

Natasha Dobrinen, University of Notre Dame

(joint work with Tom Benhamou)  The Tukey structure of ultrafilters on $\omega$ has been studied extensively in the last two decades with various works of Blass, Dobrinen, Kuzeljevic,  Mijares, Milovich, Raghavan, Shelah, Todorcevic, Trujillo, and Verner.  Research on the Galvin property for ultrafilters over uncountable cardinals, in particular on measurable cardinals, has gained recent momentum, due to applications in infinite combinatorics, cardinal arithmetic, and inner models and forcing theory,  with various works of Benhamou, Garti, Gitik, Poveda, and Shelah. Joint work with Tom Benhamou began with the observation that the Galvin property is equivalent to being not Tukey maximal; hence, Tukey types refine various Galvin properties.  We initiate the development of the Tukey theory of ultrafilters on measurable cardinals, allowing the flow of results from the countable to the uncountable and vice versa.  The situation for ultrafilters on measurable cardinals turns out to be quite different from that on $\omega$, sometimes greatly simplifying the situation on $\omega$ and sometimes posing new obstacles.  The structure of the Tukey classes also turns out to be sensitive to different large cardinal hypotheses.  We will present results from our preprint arXiv:2304.07214  and ongoing work. 

MC 5479