Meetsets

Meetsets are a concept in combinatorics, specifically within the study of set systems. A meetset of a family of sets is a subset of the underlying ground set that possesses a particular property related to the intersections of the sets in the family. More formally, for a family of sets F = {A_1, A_2, ..., A_n} where each A_i is a subset of a ground set X, a meetset M is a subset of X such that for any two sets A_i and A_j in F, the intersection A_i intersect A_j is contained within M if and only if M is contained within the intersection A_i intersect A_j. This definition can also be stated in terms of elements: an element x is in M if and only if x is in every set A_i that contains a specific element y, and this holds for all y in M. The study of meetsets often involves determining the possible sizes and structures of meetsets for various families of sets, and their relationships to other combinatorial objects like intersecting families or designs. They can arise in areas such as extremal set theory and discrete geometry.