Extremal Cases of the Ahlswede–Cai Inequality

Consider two set systems A and B in the powerset P(n) with the property that for eachA∈A there exists a uniqueB∈B such thatA⊂B. Ahlswede and Cai proved an inequality about such systems which is a generalization of the LYM and Bollobás inequalities. In this paper we characterize the structure of the extremal cases.