The number of k-intersections of an intersecting family of r-sets

The Erdős–Ko–Rado theorem tells us how large an intersecting family of r-sets from an n-set can be, while results due to Lovász and Tuza give bounds on the number of singletons that can occur as pairwise intersections of sets from such a family. sider a natural common generalization of these problems. Given an intersecting family of r-sets from an n-set and 1⩽k⩽r, how many k-sets can occur as pairwise intersections of sets from the family? For k=r and 1 this reduces to the problems described above. We answer this question exactly for all values of k and r, when n is sufficiently large. Our result is in the form of a structure theorem characterizing the extremal families in terms of extremal families for the Lovász–Tuza problem.