Abstract: The ultrafilter lemma, saying that every filter can be extended to an ultrafilter, is one of the fundamental consequences of the axiom of choice. By adding closure assumptions, and asking for extension of $\kappa$-complete filters to $\kappa$-complete ultrafilters, we obtain the notion of strongly compact cardinal, which has a very high consistency strength. 

The consistency strength of strongly compact cardinals is typically derived from the existence of uniform $\kappa$-complete ultrafilters on cardinals such as $\kappa^+$. Thus it is interesting to ask whether restricting the extension property for filters on $\kappa$, reduces the consistency strength, and maybe even allow one to get an exact equiconsistency result. 

In this talk I will show that this filter extension property still implies the failure of two successive squares, and thus it is still un-analysable by the current inner models technology (and give a wild conjecture for the consistency strength). I will also show that the extension property for normal filters on $\kappa$ and for $\kappa$-complete filters on $\kappa$ is equivalent, so it also has a very high consistency strength, answering a question of Gitik.