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 nset, to give simpler new proofs of the known asymptotic behavior of La(n, P ), as n → ∞, when P is the r-fork Vr, the four-element N poset N , and the four-element butterfly-poset B. Dedicated to Gerard Chang on the occasion of his 60th birthday 1 Largest Families Without a Forbidden Poset We study how large can a family of subsets of an n-set be that avoids containing a given finite poset P as a subposet. For two posets P = (P,≤) and P ′ = (P ′,≤′), we say that P contains P ′ as a weak subposet if there exists an injection f : P ′ → P that preserves the partial ordering, meaning that whenever u ≤′ v in P , we have f(u) ≤ f(v) in P [13]. Throughout the paper, when we say subposet, we mean weak subposet. Let [n] := {1, . . . , n}, and let the Boolean lattice Bn = (2[n],⊆) denote the power set of [n] with the set-inclusion relation. We consider families F ⊆ 2. Then F can be viewed as a subposet of Bn. If F contains no subposet P , we say F is P -free. We are interested in determining the largest size of a P -free family of subsets of [n], denoted La(n, P ). The original result of this type is Sperner’s Theorem [14] on the maximum size of antichain in the Boolean lattice. An antichain is a family of subsets such that no subset is included in another, which we can view as a poset that contains no two-element chain as a subposet. Erdős [5] generalized this to give the largest size La(n,Pk) of a family that does not contain a chain of size k, the path poset Pk consisting of k totally ordered elements A1 < A2 < · · · < Ak. Katona brought the attention of many researchers to the generalization of this problem to posets P besides chains Pk. In every case that is solved, La(n, P ) is asymptotic to a