Maximal 3-Wise Intersecting Families

Abstract

A family .7' on ground set [n] := {1, 2, ... , n} is maximal k -wise intersecting if every collection of at most k sets in .7' has non-empty intersection, and no other set can be added to .7' while maintaining this property. In 1974, Erdos and Kleitman asked for the minimum size of a maximal k-wise intersecting family. We answer their question for k = 3 and sufficiently large n. We show that the unique minimum family is obtained by partitioning the ground set [n] into two sets A and B with almost equal sizes and taking the family consisting of all the proper supersets of A and of B.