**May 18**

Miha Habič,
Charles University

**Tukey classes of complete ultrafilters**

A poset $P$ is Tukey reducible to a poset $Q$ if there is a map $f\colon Q\to P$ which takes cofinal subsets of $Q$ to cofinal subsets of $P$. The classification of all Tukey classes of posets of size continuum is not feasible, but becomes more tractable if we restrict our attention to a particular class of posets. Of particular interest are Tukey reductions between ultrafilters on $\omega$, ordered by inclusion, and it is a long-standing open question whether it is consistent that all nonprincipal ultrafilters are Tukey equivalent. I will give an overview of known results on the topic, and present some new joint work on the parallel question in the case of complete ultrafilters on uncountable cardinals.