The partition method for poset-free families

Given a finite poset P, let ${\rm La}(n,P)$ denote the largest size of a family of subsets of an n-set that does not contain P as a (weak) subposet. We employ a combinatorial method, using partitions of the collection of all full chains of subsets of the n-set, to give simpler new proofs of the known asymptotic behavior of ${\rm La}(n,P)$ , as n→∞, when P is the r-fork $\mathcal {V}_{r}$ , the four-element N poset $\mathcal {N}$ , and the four-element butterfly-poset $\mathcal {B}$ .