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Combinatorics of Finite Sets

Ian Anderson
Publisher: 
Dover Publications
Publication Date: 
2002
Number of Pages: 
272
Format: 
Paperback
Price: 
19.95
ISBN: 
9780486422572
Category: 
Textbook
BLL Rating: 

The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

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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

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