Packing Posets in the Boolean Lattice

We are interested in maximizing the number of pairwise unrelated copies of a poset P in the family of all subsets of [n]. For instance, Sperner showed that when P is one element, ( n bn2 c ) is the maximum number of copies of P . Griggs, Stahl, and Trotter have shown that when P is a chain on k elements, 1 2k−1 ( n bn2 c ) is asymptotically the maximum number of copies of P . We prove that for any P the maximum number of unrelated copies of P is asymptotic to a constant times ( n bn2 c ) . Moreover, the constant has the form 1 c(P ) , where c(P ) is the size of the smallest convex closure over all embeddings of P into the Boolean lattice.