Institute of Mathematics of the Academy of Sciences of the Czech Republic

Grant L100192251 1.7.2022 - 30.6.2024
Grantor: Czech Academy of Sciences

The main goal of the project is to establish under what conditions ultrafilters with particular combinatorial properties may or may not exist.

Recently, Dr. Cancino-Manríquez proved that consistently there is no I-ultrafilter for any ideal I with Borel complexity F sigma. As a result, J. Brendle has asked whether it is possible to generalize this theorem to get the continuum size bigger than omega 2, which is the second uncountable cardinal. Dr. Cancino-Manríquez has a partial advance in this question pointing to a positive answer: the model constructed by forcing with the Rational Perfect set forcing parametrized by a lower semicontinuous submeasure and then with the side by side product of the Silver’s forcing. This work will be continued within the framework of this project.
The second objective is to answer an open question about the existence of a model where there is no q-point but there is a rapid ultrafilter.
The third objective will be to solve completely the Isbell’s problem: whether it is consistent that all ultrafilters have maximal cofinal type. What remains open is what happens when there are no p-points.
The fourth objective addresses proposing the construction of a model where there is no q-point nor p-point. This has been a long standing question, and possibly the difficulty finds in the lack of suitable forcing techniques to produce appropriate models.

Main investigator:

Participating institutions:

Institute of Mathematics, Czech Academy of Sciences