The Erdős–Ko–Rado properties of set systems defined by double partitions

Abstract

Let F be a family of subsets of a finite set V . The star of F at v 2 V is the sub-family {A 2 F : v 2 A}. We denote the sub-family {A 2 F : |A| = r} by F(r). A double partition P of a finite set V is a partition of V into large sets that are in turn partitioned into small sets. Given such a partition, the family F(P) induced by P is the family of subsets of V whose intersection with each large set is either contained in just one small set or empty. Our main result is that, if one of the large sets is trivially partitioned (that is, into just one small set) and 2r is not greater than the least cardinality of any maximal set of F(P), then no intersecting subfamily of F(P)(r) is larger than the largest star of F(P)(r). We also characterise the cases when every extremal intersecting sub-family of F(P)(r) is a star of F(P)(r).