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 $[n]$. We prove that for any $P$ the maximum number of unrelated copies of $P$ is asymptotic to a constant times the largest binomial coefficient. Moreover, the constant has the form $\frac{1}{c(P)}$, where $c(P)$ is the size of the smallest convex closure over all embeddings of $P$ into the Boolean lattice.