  The Erdos-Ko-Rado theorem tells us how large an intersecting family of r-sets
from an n-set can be, while results due to Lovasz and Tuza give bounds on the
number of singletons that can occur as pairwise intersections of sets from such
a family.
  We consider a natural generalization of these problems. Given an intersecting
family of r-sets from an n-set and 1\leq k \leq r, how many k-sets can occur as
pairwise intersections of sets from the family? For k=r and k=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. We also characterize the extremal
families.
