Set Families with a Forbidden Subposet

We asymptotically determine the size of the largest family F of subsets of {1, . . . , n} not containing a given poset P if the Hasse diagram of P is a tree. This is a qualitative generalization of several known results including Sperner’s theorem. Introduction We say that a poset P is a subposet of a poset P ′ if there is an injective map f : P → P ′ such that a 6P b implies f(a) 6P ′ f(b). A poset P is an induced subposet of P ′ if there is an injective map f : P → P ′ for which a 6P b if and only if f(a) 6P ′ f(b). For instance, is a subposet of , but not an induced subposet. For a poset P a Hasse diagram, denoted by H(P ), is a graph whose vertices are elements of P , and xy is an edge if x < y and for no other element z of P we have x < z < y. Let [n] = {1, . . . , n}, and denote by 2 the collection of all subsets of [n]. One can think of a family F of subsets of [n] as a poset under inclusion. In this way F becomes an induced subposet of the Boolean lattice. In this paper we examine the size of the largest family F ⊂ 2 subject to the condition that F does not contain a fixed finite subposet P . We do not require P to be an induced subposet. A set family F not containing a subposet P will be called a P -free family. We denote by ex(P, n) the size of the largest P -free family F ⊂ 2. For example, the classical Sperner’s theorem [Spe28] asserts that ex( , n) = ( n ⌊n/2⌋ )