On families of subsets with a forbidden subposet

Abstract

Let F ⊂ 2[n] be a family of subsets of {1, 2,⋯, n}. For any poset H, we say is H-füree if does not contain any subposet isomorphic to H. Katona and others have investigated the behaviour of La(n, H), which denotes the maximum size of H-füree families F ⊂ 2[n]. Here we use a new approach, which is to apply methods fürom 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. © 2009 Cambridge University Press.