Combinatorics of Finite Sets

Marc Notes:
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; ... a corrected republication of the work as published by Oxford University Press, Oxford, England, and New York, in 1989 (first publication: 1987)--T.p. verso;Includes bibliographical references (p. 241-248) and index.

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

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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.