Logic Seminar
Tom Benhamou
UIC
The Galvin Property
Abstract: We present a property of filters discovered by F. Galvin which he proved to hold for normal filters over strongly regular cardinals, and gained renewed in-
terest due to recent developments in set theory. In the first part of the talk, we will provide applications of this property to infinite combintorics and to Prikry type
forcing. The second goal will be to present some strengthening of Galvin’s theorem, and prove that in some canonical inner models, every $\kappa$-complete ultrafilter over $\kappa$ has the Galvin property. We will also present constructions of filters and ultra-filters without the Galvin property and prove a new result answering a question of
Garti, Shelah and B. about the existence of such ultrafilters on very large cardinals.
In the third part of the talk, we continue the work of U.Abraham and S.Shelah
who produced a model where the club filter fails to satisfy the Galvin property in a
strong sense at $\kappa ^+$, where $\kappa$ is a regular cardinal and $2^ \kappa > \kappa ^+$.
We will produce
a model where the club filter fails to satisfy the Galvin property at $\kappa ^+$, where $\kappa$ is singular and $2^\kappa > \kappa ^+$. We will obtain this model from the optimal large cardinal
assumptions and explore the possibility of obtaining the stronger form of failure as
in the Abraham and Shelah model. This is a joint work with M.Gitik, S.Garti and
A.Poveda.
Tuesday August 30, 2022 at 4:00 PM in 636 SEO