Maximal s-Wise t-Intersecting Families of Sets: Kernels, Generating Sets, and Enumeration

A family A of k-subsets of an n-set is said to be s-wise t-intersecting if |A1∩…∩As|⩾t, for any A1, …, As∈A. For fixed s, n, k, and t, let Is(n, k, t) denote the set of all such families. A family A∈Is(n, k, t) is said to be maximal if it is not properly contained in any other family in Is(n, k, t). We show that for fixed s, k, t, there is an integer n0=n0(k, s, t), for which the maximal families in Is(n0, k, t) completely determine the maximal families in Is(n, k, t), for all n⩾n0. We give a construction for maximal families in Is(n+1, k+1, t+1) based on those in Is(n, k, t). Finally, for s=2, we classify the maximal families for k=t+1, n⩾t+2, t⩾1, and for k=t+2, n⩾t+6, t⩾1. The concepts of kernels and generating sets of a family of subsets play an important role in this work.