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Combinatorics of Finite Sets (Dover Books on Mathematics)by Ian Anderson
Synopses & ReviewsPublisher Comments:Coherent treatment provides comprehensive view of basic methods and results of the combinatorial study of finite set systems. The ClementsLindstrom extension of the KruskalKatona theorem to multisets is explored, as is the GreeneKleitman result concerning ksaturated 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. Book News Annotation:Anderson (U. of Glasgow) explores collections of subsets of a finite set where the collection is described in terms of intersection, union, or inclusive conditions. He also considers more general partially ordered sets. The 1989 edition, published by University Press, Oxford, has been slightly corrected.
Annotation c. Book News, Inc., Portland, OR (booknews.com) Synopsis:Among other subjects explored are the ClementsLindström extension of the KruskalKatona theorem to multisets and the GreeneKleitmen result concerning ksaturated chain partitions of general partially ordered sets. Includes exercises and solutions. Synopsis:Coherent treatment provides comprehensive view of basic methods and results of the combinatorial study of finite set systems. The ClementsLindstrom extension of the KruskalKatona theorem to multisets is explored, as is the GreeneKleitman result concerning ksaturated chain partitions of general partially ordered sets. Connections with Dilworth's theorem, the marriage problem, and probability are also discussed.
Table of Contents Notation
1. Introduction and Sperner's theorem 1.1 A simple intersection result 1.2 Sperner's theorem 1.3 A theorem of Bollobás Exercises 1 2. Normalized matchings and rank numbers 2.1 Sperner's proof 2.2 Systems of distinct representatives 2.3 LYM inequalities and the normalized matching property 2.4 Rank numbers: some examples Exercises 2 3. Symmetric chains 3.1 Symmetric chain decompositions 3.2 Dilworth's theorem 3.3 Symmetric chains for sets 3.4 Applications 3.5 Nested Chains 3.6 Posets with symmetric chain decompositions Exercises 3 4. Rank numbers for multisets 4.1 Unimodality and log concavity 4.2 The normalized matching property 4.3 The largest size of a rank number Exercises 4 5. Intersecting systems and the ErdösKoRado theorem 5.1 The EKR theorem 5.2 Generalizations of EKR 5.3 Intersecting antichains with large members 5.4 A probability application of EKR 5.5 Theorems of Milner and Katona 5.6 Some results related to the EKR theorem Exercises 5 6. Ideals and a lemma of Kleitman 6.1 Kleitman's lemma 6.2 The AhlswedeDaykin inequality 6.3 Applications of the FKG inequality to probability theory 6.4 Chvátal's conjecture Exercises 6 7. The KruskalKatona theorem 7.1 Order relations on subsets 7.2 The lbinomial representation of a number 7.3 The KruskalKatona theorem 7.4 Some easy consequences of KruskalKatona 7.5 Compression Exercises 7 8. Antichains 8.1 Squashed antichains 8.2 Using squashed antichains 8.3 Parameters of intersecting antichains Exercises 8 9. The generalized Macaulay theorem for multisets 9.1 The theorem of Clements and Lindström 9.2 Some corollaries 9.3 A minimization problem in coding theory 9.4 Uniqueness of a maximumsized antichains in multisets Exercises 9 10. Theorems for multisets 10.1 Intersecting families 10.2 Antichains in multisets 10.3 Intersecting antichains Exercises 10 11. The LittlewoodOfford problem 11.1 Early results 11.2 Mpart Sperner theorems 11.3 LittlewoodOfford results Exercises 11 12. Miscellaneous methods 12.1 The duality theorem of linear programming 12.2 Graphtheoretic methods 12.3 Using network flow Exercises 12 13. Lattices of antichains and saturated chain partitions 13.1 Antichains 13.2 Maximumsized antichains 13.3 Saturated chain partitions 13.4 The lattice of kunions Exercises 13 Hints and solutions; References; Index What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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