2020 seminar talk: Weakly Ramsey ultrafilters

Talk held by Noé de Rancourt (KGRC) at the KGRC seminar on 2020-04-23.

Abstract

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.

Time and Place

Talk at 4:00pm via Zoom