An Intersection-Weighted Erd\H{o}s-Ko-Rado Theorem

We consider an Erd\H{o}s-Ko-Rado type sum that weights each member of a uniform family according to its smallest intersection with the rest of the family. We prove that once the ground set is sufficiently large this sum is at most one, with equality exactly for stars. This simultaneously generalizes the usual Erd\H{o}s-Ko-Rado theorem for every intersection threshold $t$ and $n$ sufficiently large. As a consequence we also obtain an extension of Hilton's theorem on cross-intersecting families.