Weakly Ramsey ultrafilters are ultrafilters on \(\omega\) satisfying a weak local version of Ramsey's theorem; they naturally generalize Ramsey ultrafilters. It is well known that an ultrafilter on \(\omega\) is Ramsey if and only if it is minimal in the Rudin-Keisler ordering; in joint work with Jonathan Verner, we proved that similarly, weakly Ramsey ultrafilters are low in this ordering: there are no infinite chains below them. This generalizes a result of Laflamme's. In this talk, I will outline a proof of this result, and the construction of a counterexample to the converse of this fact, namely a non-weakly-Ramsey ultrafilter having exactly one Rudin-Keisler predecessor. This construction is partly based on finite combinatorics.
Weakly Ramsey ultrafilters
23.04.2020 15:00 - 16:30
Organiser:
KGRC
Location:
online via Zoom