A Group-Theoretic Setting for Some Intersecting Sperner Families

Erdos, Ko and Rado proved in 1961 that a family of pairwise intersecting /c-subsets of an n-set cannot have more members than a maximal (antichain) of /c-subsets, all of which contain a given element a, say, provided k < \n/2\. This result is closely analogous to Sperner's theorem [14] on maximal families of pairwise incomparable subsets. In fact, Bollobas [2] could generalize the Erdos-Ko-Rado Theorem, in the same spirit as the LYM-inequality generalizes Sperner's theorem. This is achieved in the following way. Associate with the family J* of subsets of the set S, its profile ( / i , . . . , /m) via