Combination of Finite Sets

By: Ian Anderson

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Paperback | 2 November 2011

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272 Pages

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21.5 x 13.5 x 1.3

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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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You Can Find This Book In

Non-Fiction Mathematics Combinatorics & Graph Theory Mathematical Foundation Set Theory

Notation
Introduction and Sperner's theorem
A simple intersection result
Sperner's theorem
A theorem of Bollob_s
Exercises 1
Normalized matchings and rank numbers
Sperner's proof
Systems of distinct representatives
LYM inequalities and the normalized matching property
Rank numbers: some examples
Exercises 2
Symmetric chains
Symmetric chain decompositions
Dilworth's theorem
Symmetric chains for sets
Applications
Nested Chains
Posets with symmetric chain decompositions
Exercises 3
Rank numbers for multisets
Unimodality and log concavity
The normalized matching property
The largest size of a rank number
Exercises 4
Intersecting systems and the Erdws-Ko-Rado theorem
The EKR theorem
Generalizations of EKR
Intersecting antichains with large members
A probability application of EKR
Theorems of Milner and Katona
Some results related to the EKR theorem
Exercises 5
Ideals and a lemma of Kleitman
Kleitman's lemma
The Ahlswede-Daykin inequality
Applications of the FKG inequality to probability theory
Chv_tal's conjecture

The Kruskal-Katona theorem
Order relations on subsets
The l-binomial representation of a number
The Kruskal-Katona theorem
Some easy consequences of Kruskal-Katona
Compression
Exercises 7
Antichains
Squashed antichains
Using squashed antichains
Parameters of intersecting antichains
Exercises 8
The generalized Macaulay theorem for multisets
The theorem of Clements and Lindstrwm
Some corollaries
A minimization problem in coding theory
Uniqueness of a maximum-sized antichains in multisets
Exercises 9
Theorems for multisets
Intersecting families
Antichains in multisets
Intersecting antichains
Exercises 10
The Littlewood-Offord problem
Early results
M-part Sperner theorems
Littlewood-Offord results
Exercises 11
Miscellaneous methods
The duality theorem of linear programming
Graph-theoretic methods
Using network flow
Exercises 12
Lattices of antichains and saturated chain partitions
Antichains
Maximum-sized antichains
Saturated chain partitions
The lattice of k-unions
Exercises 13
Hints and solutions
References
Index
Table of Contents provided by Publisher. All Rights Reserved.

Combination of Finite Sets

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