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Combinatorics of finite sets

Author: Ian Anderson, Ph. D.
Publisher: Mineola, N.Y. : Dover Publications, 2002.
Series: Dover books on mathematics.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
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.  Read more...
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Details

Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Ian Anderson, Ph. D.
ISBN: 0486422577 9780486422572
OCLC Number: 48907086
Notes: " ... a corrected republication of the work as published by Oxford University Press, Oxford, England, and New York, in 1989 (first publication: 1987)"--Title page verso.
Description: xv, 250 pages ; 22 cm.
Contents: Machine generated contents note: 1. Introduction and Sperner's theorem --
1.1 A simple intersection result --
1.2 Sperner's theorem --
1.3 A theorem of Bollobas --
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 Erdos-Ko-Rado --
theorem --
5.1 The EKR theorem --
5.2 Generalizations of EKR --
5.3 Intersecting aintichains 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 Ahlswede-Daykin inequality --
6.3 Applications of the FKG inequality to probability --
theory --
6.4 Chvatal's conjecture --
Exercises 6 --
7. The Kruskal-Katona theorem --
7.1 Order relations on subsets --
7.2 The i-binomial representation of a number --
7.3 The Kruskal-Katona theorem --
7.4 Some easy consequences of Kruskal-Katona --
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 Lindstrom --
9.2 Some corollaries --
9.3 A minimization problem in coding theory --
9.4 Uniqueness of maximum-sized 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 Littlewood-Offord problem --
11.1 Early results --
11.2 M-part Sperner theorems --
11.3 Littlewood-Offord results --
Exercises 11 --
12. Miscellaneous methods --
12.1 The duality theorem of linear programming --
12.2 Graph-theoretic methods --
12.3 Using network flow --
Exercises 12 --
13. Lattices of antichains and saturated chain partitions --
13.1 Antichains --
13.2 Maximum-sized antichains --
13.3 Saturated chain partitions --
13.4 The lattice of k-unions --
Exercises 13 --
Hints and solutions --
References --
Index.
Series Title: Dover books on mathematics.
Responsibility: Ian Anderson.
More information:

Abstract:

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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