Design Theory: (6 Discrete Mathematics and Its Applications)

This is an ideal textbook for advanced undergraduate and graduate courses in Combinatorial Design Theory. The text Features clear explanations of basic designs such as Steiner and Kirkman triple systems, mutually orthogonal Latin squares, finite projective and affine planes, and Steiner quadruple systems. Design Theory offers a progressive approach to the subject, with carefully ordered results. It begins with simple constructions that gradually increase in complexity. Each design has a construction that contains new ideas, or that reinforces and builds upon similar ideas previously introduced. The many illustrations aid in understanding and enjoying the application of the constructions described. .

Table of Contents:
Steiner Triple Systems The Existence Problem u º 3 (mod 6): The Bose Construction u º 1 (mod 6): The Skolem Construction u º 5 (mod 6): The 6n + 5 Construction Quasigroups with Holes and Steiner Triple Systems Constructing Quasigroups with Holes Constructing Steiner Triple Systems using Quasigroups with Holes The Wilson Construction Cyclic Steiner Triple Systems l-Fold Triple Systems Triple Systems of Index l > 1 The Existence of Idempotent Latin Squares 2-Fold Triple Systems Constructing 2-Fold Triple Systems l = 3 and 6 l-Fold Triple Systems in General Maximum Packings and Minimum Coverings The General Problem Maximum Packings Minimum Coverings Kirkman Triple Systems A Recursive Construction Constructing Pairwise Balanced Designs Mutually Orthogonal Latin Squares Introduction The Euler and MacNeish Conjectures Disproof of the MacNeish Conjecture Disproof of the Euler Conjecture Orthogonal Latin Squares of Order n º 2 (mod 4) Affine and Projective Planes Affine Planes Projective Planes Connections between Affine and Projective Planes Connections between Affine Planes and Complete Sets of MOLS (n) Coordinating the Affine Planes Steiner Quadruple Systems Introduction Constructions of Steiner Quadruple Systems The Stern and Lenz Lemma The (3u - 2u)-Construction Appendices A. Cyclic Steiner Triple Systems B. Answers to Selected Exercises Index

Review :
Design Theory is exploring an area in combinatorics that concerns the conditions for existence of several important block designs and methods to construct them. It explains these techniques in good detail and accompanies the theory with helpful illustrations that serve their purpose admirably. The material of this book has been carefully ordered and results, theorems and constructions build successively on previously presented definitions and explanations … a helpful addition to any combinatorist’s library and provides a nice introduction to block design construction techniques, with a collection of carefully selected and effectively presented topics. —Dimitris Papamichail, SIGACT News, December 2011