Maximal intersection critical families of finite sets Original Research Article

A finite family of pairwise intersecting r-sets is a maximal r-clique if it cannot be extended to another r-clique by adding a new r-set. It is intersection critical if it is not possible to replace any edge by some of its proper subsets, without violating the intersection property. We prove that if a maximal r-clique H, distinct from Kr+1r is not intersection critical, then | H | > | V (H) |. Moreover, we prove that the system of lines of a projective plane not passing through a fixed point is an intersection critical r-clique, not contained in any larger one.