On Families of Subsets With a Forbidden Subposet

Let F ⊂ 2[n] be a family of subsets of {1, 2, . . . , n}. For any poset H, we say F is H-free if F does not contain any subposet isomorphic to H. Katona and others have investigated the behavior of La(n,H), which denotes the maximum size of H-free families F ⊂ 2[n]. Here we use a new approach, which is to apply methods from extremal graph theory and probability theory to identify new classes of posets H, for which La(n,H) can be determined asymptotically as n →∞ for various posets H, including two-end-forks, up-down trees, and cycles C4k on two levels. Dedicated to Prof. William T. Trotter on the occasion of his 65th birthday