Hilton–Milner theorem

The Hilton–Milner theorem is a result in combinatorics that describes the largest possible size of a family of sets where no one set is contained within another (an *intersecting family*), excluding the case where all sets share a common element. Essentially, it identifies the maximum number of such sets you can have when they are all intersecting but don’t share a single universal element, providing a precise boundary for the size of these complex set families. This helps mathematicians understand the limits of how large certain intersecting collections can be under these constraints.