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Combinatorics of Finite Sets (Dover Books on Mathematics)

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Combinatorics of Finite Sets (Dover Books on Mathematics) Cover

 

Synopses & Reviews

Publisher Comments:

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.

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 Clements-Lindström extension of the Kruskal-Katona theorem to multisets and the Greene-Kleitmen result concerning k-saturated 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 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.

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 Erdös-Ko-Rado 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 Ahlswede-Daykin inequality

  6.3 Applications of the FKG inequality to probability theory

  6.4 Chvátal's conjecture

    Exercises 6

7. The Kruskal-Katona theorem

  7.1 Order relations on subsets

  7.2 The l-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 Lindström

  9.2 Some corollaries

  9.3 A minimization problem in coding theory

  9.4 Uniqueness of a 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

Product Details

ISBN:
9780486422572
Author:
Anderson, Ian
Publisher:
Dover Publications
Author:
Mathematics
Location:
Mineola, N.Y.
Subject:
Combinatorial analysis
Subject:
Set Theory
Subject:
Combinatorics
Subject:
General Mathematics
Subject:
Discrete Mathematics
Subject:
Finite Sets
Subject:
Combinatorics of finite sets
Subject:
Combinatorics of finite s
Subject:
ETS
Subject:
Mathematics : Set Theory
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Series Volume:
1737-17
Publication Date:
20111131
Binding:
TRADE PAPER
Language:
English
Pages:
272
Dimensions:
8.5 x 5.38 in 0.67 lb

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Related Subjects

Science and Mathematics » Mathematics » Combinatorics
Science and Mathematics » Mathematics » Set Theory

Combinatorics of Finite Sets (Dover Books on Mathematics) New Trade Paper
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Product details 272 pages Dover Publications - English 9780486422572 Reviews:
"Synopsis" by ,
Among other subjects explored are the Clements-Lindström extension of the Kruskal-Katona theorem to multisets and the Greene-Kleitmen result concerning k-saturated chain partitions of general partially ordered sets. Includes exercises and solutions.

"Synopsis" by , 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.
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