3 problems

Menas's conjecture. Not every strongly compact cardinal has a function with the Menas property. Menas proved that every strongly compact limit of strongly compact cardinals has suc…

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…

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…