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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