The Width of Random Subsets of Boolean Lattices

Suppose we toss an independent coin with probability of success p for each subset of [n]={1,…,n}, and form the random hypergraph P(n,p) by taking as hyperedges the subsets with successful coin tosses. We investigate the cardinality of the largest Sperner family contained in P(n,p). We obtain a sharp result for the range of p=p(n) in which this Sperner family has cardinality comparable to the cardinality of P(n,p).