## Combinatorics: Proceedings of the NATO Advanced Study Institute, Held at Nijenrode Castle, Breukelen, The Netherlands, 8-20 July 1974, Volume 1Combinatorics 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. |

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### Contents

Constructions and Uses of Pairwise Balanced Designs | 19 |

On Transversal Designs | 43 |

On Finite NonCommutative Affine Spaces | 65 |

Coding Theory | 118 |

Weight Enumerators of Codes 115 | 141 |

Recent Results on Perfect Codes and Related Topics | 163 |

Irreducible Cyclic Codes and Gauss Sums | 185 |

Isomorphism Problems for Hypergraphs | 205 |

Foundations Partitions and Combinatorial Geometry | 261 |

Sperner Families and Partitions of a Partially Ordered | 291 |

Combinatorial Reciprocity Theorems | 307 |

Difference Sets | 321 |

Invariant Relations Coherent Configurations | 347 |

413 | |

Suborbits in Transitive Permutation Groups | 419 |

Groups Polar Spaces and Related Structures | 451 |

On an Extremal Property of Antichains in Partial | 243 |

Applications of Ramsey Style Theorems to Eigenvalues | 245 |

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### Common terms and phrases

2-transitive abelian adjacent affine space algebra antichains association scheme assume automorphism group cliques Combinatorial Theory condition contains defined DEMBOWSKI denote difference set disjoint divisor doubly transitive edges elements equation equivalent ERDOS example exists finite nearaffine space fixed G is transitive Gauss sum geometry given graph group G Hadamard Hadamard matrix Hence hypergraph hyperplane implies incidence matrix intersection invariant irreducible isomorphic k-families KANTOR KLEITMAN lattice lemma Let G linear Math maximal N.J.A. SLOANE normal subgroup obtain orbits pairs parameters partially ordered partition perfect codes permutation groups polar space polynomial positive integers prepolar space primitive problem projective space proposition proved RAMSEY's theorem rank relation result satisfying self-dual code Sperner Sperner's theorem straight lines subgraph subgroup suborbit subsets subspace Suppose symmetric design t-designs theorem 2.1 transitive groups vector vertex vertices weight enumerator WIELANDT x,y e xLJy

### Popular passages

Page 417 - Ward, On Ree's series of simple groups, Trans. Amer. Math. Soc. 121 (1966), 62-89. MR 33 #5752.