Theorem of Erdős–Ko–Rado

The Erdős–Ko–Rado theorem addresses how large a family of sets can be when every pair of sets shares at least one common element. Imagine you have a collection of groups, each containing a fixed number of items, and you want all these groups to always have some item in common. The theorem states that the maximum size of such a collection is achieved when all groups include a specific, fixed item. This result precisely characterizes the largest possible family of intersecting sets under these conditions, providing deep insight into combinatorial structures and their limitations.