# 2019 seminar talk: Strong compactness and the filter extension property

Talk held by Yair Hayut (KGRC) at the KGRC seminar on 2019-03-21.

### Abstract

The notion of strongly compact cardinal is one of the earliest large cardinal axioms, yet it is still poorly understood.

I will review some classical and semi-classical connections between partial strong compactness, the strong tree property and the filter extension property, getting a level-by-level equivalence and an elementary embedding characterization.

This analysis is especially interesting for the property "every $\kappa$-complete filter on $\kappa$ can be extended to a $\kappa$-complete ultrafilter" (where $\kappa$ is uncountable). This property was isolated by Mitchell and was named "$\kappa$-compactness" by Gitik. In his recent paper, Gitik showed that some definable versions of it have a relatively low consistency strength, yet others provide an inner models with a Woodin cardinal. Applying the equivalence above to this case, I will improve the previously known lower bound for $\kappa$-compactness.

Then, I'll move to a more speculative area, and conjecture that $\kappa$-compactness is equiconsistent with a certain large cardinal axiom in the realm of subcompact cardinals. I will give a few arguments in favour of this conjecture.

A video recording of this talk is available on YouTube.

Slides for this talk are available here.