Intersection properties of boxes part II: Extremal families

In a previous paper (this Journal, Vol. 62, pp. 283–301) we proved an Upper-Bound Theorem for finite families of boxes inR ^d with edges parallel to the coordinate axes. This theorem concerns the maximum possible numbers of intersecting subfamilies of a family having a given number of members and a given clique number. Here we give an intrinsic characterization ofextremal families of boxes, i.e., families for which all the respective maximum numbers are achieved. We also deal with the problem of enumerating the possible intersection types of extremal families.