Combinatorial Theory

Cover......Page 1
Title......Page 2
Wiley-Interscience series in Discrete mathematics......Page 5
Dedication......Page 6
Preface......Page 7
Preface to the First Edition......Page 8
Contents......Page 10
1.1 Definitions......Page 17
1.2 Applications to Probability......Page 20
Problems......Page 22
2.1 The Principle of Inclusion and Exclusion. Mdbius Inversion......Page 24
2.2 Partially Ordered Sets and Their Mobius Functions......Page 31
Problems......Page 34
3.1 Rules and Properties......Page 36
3.2 Combinatorial Problems......Page 40
Problems......Page 44
4.1 Partitions. Identities and Arithmetic Properties......Page 47
4.2 Asymptotic Properties of p(n)......Page 59
Problems......Page 62
5.1 The Theorems of P.Hall and D.Konig......Page 64
5.2 The Permanent......Page 72
5.3 Proof of the van der Waerden Conjecture......Page 74
5.4 Permanents of Integral Matrices with Constant Line Sum......Page 85
Problems......Page 88
6.1 Statement of the Theorem......Page 89
6.2 Application of Ramsey\'s Theorem......Page 90
Problems......Page 91
7.1 The Assignment Problem......Page 93
7.2 Dilworth\'s Theorem......Page 97
Problems......Page 100
8.1 Convex Spaces. Convex Cones and Their Duals......Page 101
8.2 Linear Inequalities......Page 105
8.3 Linear Programming. The Simplex Method......Page 112
9.1 Complete Cycles......Page 126
9.2 Theorems on Graphs......Page 128
9.3 Proof of the DeBruijn Theorem......Page 130
9.4 Strongly Regular Graphs......Page 134
9.5 Finite Permutation Groups of Rank 3......Page 138
10.1 General Discussion......Page 142
10.2 Elementary Theorems on Block Designs......Page 145
10.3 The Bruck-Ryser-Chowla Theorem......Page 149
10.4 Statement of the Hasse-Minkowski Theorem. Applications......Page 155
11.1 Examples and Definitions......Page 163
11.2 Finite Fields......Page 166
11.3 The Theorem of Singer......Page 171
11.4 The Multiplier Theorem......Page 175
11.5 Difference Sets in General Groups......Page 180
11.6 Some Families of Difference Sets......Page 186
12.1 Foundations......Page 215
12.2 Finite Geometries as Block Designs......Page 219
12.3 Finite Planes......Page 221
12.4 Some Types of Finite Planes......Page 227
13.1 Orthogonality and Orthogonal Arrays......Page 238
13.2 Main Theorems......Page 239
13.3 Constructions of Orthogonal Squares......Page 244
13.4 The End of the Euler Conjecture......Page 250
14.1 Paley\'s Constructions......Page 254
14.2 Williamson\'s Method......Page 270
14.3 An Infinite Class of Williamson Matrices......Page 273
14.4 Three Recent Methods......Page 277
15.2 Basic Definitions. The Hanani Theorems......Page 280
15.3 Direct Construction Methods......Page 287
15.4 Triple Systems......Page 293
15.5 Block Designs with k Greater Than 3......Page 305
15.6 Wilson\'s Theorem......Page 309
15.7 Some Infinite Families of Designs......Page 321
15.8 Biplanes......Page 336
16.1 Connor\'s Method......Page 352
16.2 Copositive and Completely Positive Quadratic Forms......Page 364
16.3 Rational Completions of Incidence Matrices......Page 375
16.4 Integral Solutions of the Incidence Equation......Page 384
17.1 Error Correcting Codes......Page 392
17.2 Weight Enumerators. The Mac Williams Equations......Page 393
17.3 Applications of Codes to Designs. General Theory......Page 397
17.4 Group Invariants. Gleason\'s Theorem and Its Generalizations......Page 402
17.5 Applications to Planes of Order 10......Page 406
17.6 The Symmetric (41, 16, 6) Design......Page 415
Appendix I Balanced Incomplete Block Designs with from 3 to 20 Replications......Page 421
Appendix II Hadamard Matrices of the Williamson Type......Page 440
BIBLIOGRAPHY......Page 444
INDEX......Page 453