The partition method for poset‐free families

Given a finite poset P, let $La(n,P)$ 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 $La(n,P)$ , as n→∞, when P is the r‐fork ${𝒱}_r$ , the four‐element N poset $𝒩$ , and the four‐element butterfly‐poset $ℬ$ .