Transversal Theory

Name: Transversal Theory | An Account of Some Aspects of Combinatorial Mathematics
Brand: Academic Press
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An Account of Some Aspects of Combinatorial Mathematics

Mirsky(Autor*in)

Mirsky(Herausgeber*in)

Academic Press

Erschienen am 14. Mai 2014

271 Seiten

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978-0-08-095584-1 (ISBN)

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Inhalt

Front Cover
Transversal Theory: An account of some aspects of combinatorial mathematics
Copyright Page
Contents
Preface
Chapter 1. Sets, Topological Spaces, Graphs
1.1 Sets and mappings
1.2 Families
1.3 Mapping theorems and cardinal numbers
1.4 Boolean atoms
1.5 The lemmas of Zorn and Tukey
1.6 Tychonoff's theorem
1.7 Graphs
Notes on Chapter 1
Chapter 2. Hall's Theorem and the Notion of Duality
2.1 Transversals, representatives, and representing sets
2.2 Proofs of the fundamental theorem for finite families
2.3 Duality
Notes on Chapter 2
Chapter 3. The Method of 'Elementary Constructions'
3.1 'Elementary constructions'
3.2 Transversal index
3.3 Further extensions of Hall's theorem
3.4 A self-dual variant of Hall's theorem
Notes on Chapter 3
Chapter 4. Rado's Selection Principle
4.1 Proofs of the selection principle
4.2 Transfinite form of Hall's theorem
4.3 A theorem of Rado and Jung
4.4 Dilworth's decomposition theorem
4.5 Miscellaneous applications of the selection principle
Notes on Chapter 4
Chapter 5. Variants, Refinements, and Applications of Hall's Theorem
5.1 Disjoint partial transversals
5.2 Strict systems of distinct representatives
5.3 Latin rectangles
5.4 Subsets with a prescribed pattern of overlaps
Notes on Chapter 5
Chapter 6. Independent Transversals
6.1 Pre-independence and independence
6.2 Rado's theorem on independent transversals
6.3 A characteristic property of independence structures
6.4 Finite independent partial transversals
6.5 Transversal structures and independence structures
6.6 Marginal elements
6.7 Axiomatic treatment of the rank function
Notes on Chapter 6
Chapter 7. Independence Structures and Linear Structures
7.1 A hierarchy of structures
7.2 Bases of independence spaces
7.3 Totally admissible sets
7.4 Set-theoretic models of independence structures
Notes on Chapter 7
Chapter 8. The Rank Formula of Nash-Williams
8.1 Sums of independence structures
8.2 Disjoint independent sets
8.3 A characterization of transversal structures
8.4 Symmetrized form of Rado's theorem on independent transversals
Notes on Chapter 8
Chapter 9. Links of Two Finite Families
9.1 The notion of a link
9.2 Common representatives
9.3 The criterion of Ford and Fulkerson
9.4 Common representatives with restricted frequencies
9.5 An insertion theorem for common transversals
9.6 Harder results for a single family
Notes on Chapter 9
Chapter 10. Links of Two Arbitrary Families
10.1 The theorem of Mendelsohn and Dulmage and its interpretations
10.2 Systems of representatives with repetition
10.3 Common systems of representatives with defect
10.4 Common transversals of two families
10.5 Common transversals of maximal subfamilies
Notes on Chapter 10
Chapter 11. Combinatorial Properties of Matrices
11.1 The language of matrix theory
11.2 Theorems of König, Frobenius, and Rado
11.3 Diagonals of doubly-stochastic matrices
11.4 Doubly-stochastic patterns
11.5 Existence theorems for integral matrices
Notes on Chapter 11
Chapter 12. Conclusion
12.1 Current trends in transversal theory
12.2 Future research and open questions
Miscellaneous Exercises
Bibliography
Index of Symbols
Index of Authors
General Index
Mathematics in Science and Engineering

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