Packing Posets in the Boolean Lattice

Abstract

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, ${\dbinom {n}{\lfloor \frac {n}{2}\rfloor }}$ is the maximum number of copies of P. Griggs, Stahl, and Trotter have shown that when P is a chain on k elements, $\frac {1}{2^{k-1}}{\dbinom {n}{\lfloor \frac {n}{2}\rfloor }}$ 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 ${\dbinom {n}{\lfloor \frac {n}{2}\rfloor }}$. 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.