Combinatorics : proceedings of the NATO Advanced Study Institute, held at Nijenrode Castle, Breukelen, the Netherlands, 8-20 July 1974

Combinatorics has come of age. It had its beginnings in a number of puzzles which have still not lost their charm. Among these are EULER'S problem of the 36 officers and the KONIGSBERG bridge problem, BACHET's problem of the weights, and the Reverend T.P. KIRKMAN'S problem of the schoolgirls. Many of the topics treated in ROUSE BALL'S Recreational Mathe­ matics belong to combinatorial theory. All of this has now changed. The solution of the puzzles has led to a large and sophisticated theory with many complex ramifications. And it seems probable that the four color problem will only be solved in terms of as yet undiscovered deep results in graph theory. Combinatorics and the theory of numbers have much in common. In both theories there are many prob­ lems which are easy to state in terms understandable by the layman, but whose solution depends on complicated and abstruse methods. And there are now interconnections between these theories in terms of which each enriches the other. Combinatorics includes a diversity of topics which do however have interrelations in superficially unexpected ways. The instructional lectures included in these proceedings have been divided into six major areas: 1. Theory of designs; 2. Graph theory; 3. Combinatorial group theory; 4. Finite geometry; 5. Foundations, partitions and combinatorial geometry; 6. Coding theory. They are designed to give an overview of the classical foundations of the subjects treated and also some indication of the present frontiers of research. Front Matter....Pages I-VIII Front Matter....Pages 1-1 Indeterminates and Incidence Matrices....Pages 3-17 Constructions and Uses of Pairwise Balanced Designs....Pages 19-42 On Transversal Designs....Pages 43-53 Combinatorics of Finite Geometries....Pages 55-63 On Finite Non-Commutative Affine Spaces....Pages 65-113 Weight Enumerators of Codes....Pages 115-142 The Association Schemes of Coding Theory....Pages 143-161 Recent Results on Perfect Codes and Related Topics....Pages 163-183 Irreducible Cyclic Codes and Gauss Sums....Pages 185-202 Front Matter....Pages 203-203 Isomorphism Problems for Hypergraphs....Pages 205-214 Extremal Problems for Hypergraphs....Pages 215-244 Applications of Ramsey Style Theorems to Eigenvalues of Graphs....Pages 245-259 Some Recent Developments in Ramsey Theory....Pages 261-276 On An Extremal Property of Antichains in Partial Orders. The Lym Property and Some of Its Implications and Applications....Pages 277-290 Sperner Families and Partitions of A Partially Ordered Set....Pages 291-306 Combinatorial Reciprocity Theorems....Pages 307-318 Front Matter....Pages 319-319 Difference Sets....Pages 321-346 Invariant Relations, Coherent Configurations and Generalized Polygons....Pages 347-363 2-Transitive Designs....Pages 365-418 Suborbits in Transitive Permutation Groups....Pages 419-450 Front Matter....Pages 319-319 Groups, Polar Spaces and Related Structures....Pages 451-482 Proceedings of the NATO Advanced Study Institute, Breukelen, The Netherlands, July 8-30, 1974