4 problems

Let $d6ff$ be an inaccessible cardinal. Suppose there is a $d70f$-complete ultrafilter extending the club filter on $P_d70f(d6ff)$. Supercompactness characterization conjecture. Th…

Let $d708$ be a strongly compact cardinal and let $d707$ be a supercompact cardinal. Solovay's conjecture. Every strongly compact cardinal is supercompact. Keisler–Tarski asked whe…

Assume the Ultrapower Axiom. Let $\delta$ be a regular cardinal and let $\kappa\leq\delta$ be the least $\delta$-strongly compact cardinal. The $<_S$-least uniform countably comple…

Let $\tau$geq$omega$ be a cardinal. A cardinal is strongly compact if every set $A$ has a fine $\tau$-complete ultrafilter on $P_\tau(A)$, and it is supercompact if every set $A$ h…