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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.
Sommario
1.- Theory of Designs.- Indeterminates and Incidence Matrices.- Constructions and Uses of Pairwise Balanced Designs.- On Transversal Designs.- Finite Geometry.- Combinatorics of Finite Geometries.- On Finite Non-Commutative Affine Spaces.- Coding Theory.- Weight Enumerators of Codes.- The Association Schemes of Coding Theory.- Recent Results on Perfect Codes and Related Topics.- Irreducible Cyclic Codes and Gauss Sums.- 2.- Graph Theory.- Isomorphism Problems for Hypergraphs.- Extremal Problems for Hypergraphs.- Applications of Ramsey Style Theorems to Eigenvalues of Graphs.- Foundations, Partitions and Combinatorial Geometry.- Some Recent Developments in Ramsey Theory.- On an Extremal Property of Antichains in Partial orders. The LYM Property and Some of Its Implications and Applications.- Sperner Families and Partitions of a Partially Ordered Set.- Combinatorial Reciprocity Theorems.- 3.- Combinatorial Group Theory.- Difference Sets.- Invariant Relations, Coherent Configurations and Generalized Polygons.- 2-Transitive Designs.- Suborbits in Transitive Permutation Groups.- Groups, Polar Spaces and Related Structures.
