A Family of Inequalities for Intersecting Antichain of Subsets of an n-set

There are two seemingly unrelated inequalities for the intersecting antichain of subsets of ann-element set. One was obtained by Bollobلs in 1973 and the other was obtained by Greene, Katona and Kleitman in 1976. By introducing a new family of non-negative real valued set functions and applying cyclic permutation approach, a family of more general inequalities is obtained. The left-hand sides of the inequalities of Bollobلs and Greeneet al.are just the two extreme values of a linear function which appears in the left-hand side of one of our special family of inequalities. Furthermore, some new inequalities are provided.