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Let P be a poset, consisting of all sets X subset of or equal to [n] = {1, 2, ..., n} which contain at least one of a given collection F of 2-subsets of [n], ordered by inclusion. By modifying a construction of Greene and Kleitman, we show that if F is hamiltonian, that is, contains {1, 2}, {2, 3}, ...
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Let F be a nonempty collection of subsets of [n] = {1, 2,..., n}, each having cardinality t. Denote by P-F the poser consisting of all subsets of [n] which contain at least one member of F, ordered by set-theoretic inclusion. In 1980, K. W. Lih conjectured that P-F has the Sperner property for all 1...
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A set X of vertices of a graph is said to be dependent if X is not an independent set. For the graph G, let P-k(G) denote the set of dependent sets of cardinality k. In this paper, we show that if G is a connected graph on 2n vertices where n greater than or equal to 3 then \P-n(G)\ greater than or ...
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A set of vertices in a graph is said to be dependent if it is not independent. Let p(k)(G) denote the number of dependent sets of size k in the graph G. We show that, for any graph G, the sequence {p(k)(G)} is logarithmically concave. (C) 2001 Elsevier Science.