Combinatorics of Finite Sets
Coherent treatment provides comprehensive view of basic methods and results of the combinatorial study of finite set systems. The Clements-Lindstrom extension of the Kruskal-Katona theorem to multisets is explored, as is the Greene-Kleitman result concerning k-saturated chain partitions of general partially ordered sets. Connections with Dilworth's theorem, the marriage problem, and probability are also discussed. Each chapter ends with a helpful series of exercises and outline solutions appear at the end. "An excellent text for a topics course in discrete mathematics." — Bulletin of the American Mathematical Society.
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Ajrf antichain of subsets antichain s& basic elements binomial coefficients blocks chain partition collection of subsets consider consists contain contradicting Corollary corresponding cyclic permutation Daykin define denote the number denote the set Dilworth's theorem disjoint distributive lattice divisors downset EKR theorem elements at level example Exercise exists fc-sets fc-union fc-vectors following theorem given Greene and Kleitman Griggs induction hypothesis integer intersecting antichain intersecting family Kruskal-Katona theorem l)-sets l)-subsets lattice least Lemma Let jrf Let s& lexicographic order log concave LYM inequality marriage theorem maximal elements maximum maximum-sized antichain members of s& multisets n-set n-tuples non-negative normalized matching property Note number of sets obtain pairs permutation poset of divisors proof of Theorem ranked poset replace result S(kl sequence set of subsets sets in s& Sperner's theorem squashed antichain squashed order symmetric chain decomposition union vectors