Will give a Logic Seminar entitled:
As many $Q$-points as you would like
Abstract: Ramsey ultrafilters are those ultrafilters that give witnesses to Ramsey's theorem. For ultrafilters, the property of being Ramsey is equivalent to the conjunction of two other properties: being a $P$-point and a $Q$-point. Consistently, both of these properties are strictly weaker than being Ramsey. It is a result of Shelah (1982) that, consistently, there may be exactly $n$ many Ramsey ultrafilters up to isomorphism, for any positive natural number $n$. Later, again Shelah (1998) constructed a model in which there is a unique $P$-point. This left the question of whether it is consistent to have good control on the number of $Q$-points as well. Twenty-six years after this, Mildenberger (2024) constructed a model with exactly two and another model with exactly three near coherence classes of ultrafilters. By the nature of her models, they contained a unique $Q$-point and exactly two $Q$-points, respectively. In a recent preprint, Halbeisen, Horvath and Shelah (2025) constructed a model with a unique $Q$-point, and $2^{2^{\aleph_0}}$ many near coherence classes. Their method was to iteratively pseudo-intersect a Ramsey ultrafilter in the ground model. In this talk, we will present the consistency of there being exactly $n$ many $Q$-points for any positive natural number $n$, which is accomplished by utilizing a higher dimensional variant of Mathias forcing. Furthermore, we will show that if one replaces Mildenberger's Matet forcing restricted to a Matet-adequate family by its Matet-Mathias variant, then one still gets a model with exactly two $Q$-points, but this time with $2^{2^{\aleph_0}}$ many near coherence classes. This is joint work with Lorenz Halbeisen and Silvan Horvath.
Date: 09-23-2025
Time: 2:00 pm
Location: 125 Hayes-Healy Bldg
Originally published at math.nd.edu.