Let us recall that given two partial orders $P$ and $Q$, a function $f:P \to Q$ is cofinal
whenever it sends cofinal subsets of $P$ to cofinal subsets of $Q$. A partial order $Q$ is Tukey
reducible to a partial order $P$ (or $Q$ is Tukey below $P$) if there is a cofinal function from $P$ to $Q$.
In this setting, ultrafilters on the natural numbers can be seen as partial orders with the inverse
set inclusion relation, and the Tukey reducibility relation induces a preorder on the family of
non-principal ultrafilters on $\omega$. It is well known that ZFC proves the existence of ultrafilters
which are maximal in this preorder; such ultrafilters are known as Tukey top ultrafilters. It is an
old problem of J. R. Isbell, dating back to 1965, to answer the question about the existence of
non-Tukey top ultrafilters, and many consistency results in the affirmative are known
(for example, p-points are not Tukey top, and more generally, basically generated ultrafilters
are not Tukey top ultrafilters). We will present the two recent results: 1) the consistent non-existence
of basically generated ultrafilters; 2) the consistency of all ultrafilters being Tukey top.
This is joint work with J. Zapletal.