**Design Theory, Second Edition** presents some of the most important techniques used for constructing combinatorial designs. It augments the descriptions of the constructions with many figures to help students understand and enjoy this branch of mathematics.

This edition now offers a thorough development of the embedding of Latin squares and combinatorial designs. It also presents some pure mathematical ideas, including connections between universal algebra and graph designs.

The authors focus on several basic designs, including Steiner triple systems, Latin squares, and finite projective and affine planes. They produce these designs using flexible constructions and then add interesting properties that may be required, such as resolvability, embeddings, and orthogonality. The authors also construct more complicated structures, such as Steiner quadruple systems.

By providing both classical and state-of-the-art construction techniques, this book enables students to produce many other types of designs.

- 1 Steiner Triple Systems
- 1.1 The existence problem
- 1.2 The Bose Construction
- 1.3 The Skolem Construction
- 1.4 The 6n + 5 Construction + 5 Construction
- 1.5 Quasigroups with holes and Steiner triple systems : The 6n + 5 Construction + 5 Construction
- 1.5.1 Constructing quasigroups with holes
- 1.5.2 Constructing Steiner triple systems using quasigroupswith holes
- 1.6 The Wilson Construction
- 1.7 Cyclic Steiner triple systems
- 1.8 The 2n + 1 and 2 + 1 and 2n + 7 Constructions + 7 Constructions
- 2 λ-Fold Triple Systems λ-Fold Triple Systems
- 2.1 Triple systems of index λ > 1 Triple systems of index λ > 1
- 2.2 The existence of idempotent latin squares
- 2.3 2-Fold triple systems
- 2.3.1 Constructing2-foldtriple systems
- 2.4 Mendelsohn triple systems
- 2.5 λ = 3 and 6 λ = 3 and 6
- 2.6 λ-Fold triple systems in general λ-Fold triple systems in general
- 3 Quasigroup Identities and Graph Decompositions
- 3.1 Quasigroup identities
- 3.2 Mendelsohntriple systems revisited
- 3.3 Steiner triple systems revisited
- 4 Maximum Packings and Minimum Coverings
- 4.1 The general problem
- 4.2 Maximum packings
- 4.3 Minimum coverings
- 5 Kirkman Triple Systems
- 5.1 A recursive construction
- 5.2 Constructing pairwise balanced designs
- 6 Mutually Orthogonal Latin Squares
- 6.1 Introduction
- 6.2 The Euler and MacNeish Conjectures
- 6.3 Disproof of the MacNeish Conjecture
- 6.4 Disproof of the Euler Conjecture
- 6.5 Orthogonal latin squares of order n ≡ 2(mod 4) ≡ 2(mod 4)
- 7 Affine and Projective Planes
- 7.1 Affine planes
- 7.2 Projective planes
- 7.3 Connections between affine and projective planes
- 7.4 Connection between affine planes and complete sets of MOLS
- 7.5 Coordinatizing the affine plane
- 8 Intersections of Steiner Triple System
- 8.1 Teirlinck’s Algorithm Teirlinck’s Algorithm
- 8.2 Thegeneral intersectionproblem
- 9 Embeddings
- 9.1 Embedding latin rectangles‐necessary conditions Embedding latin rectangles‐necessary conditions
- 9.2 Edge‐coloring bipartite graphs Edge‐coloring bipartite graphs
- 9.3 Embedding latin rectangles: Ryser’s Sufficient Conditions Embedding latin rectangles: Ryser’s Sufficient Conditions
- 9.4 Embedding idempotent commutative latin squares: Cruse’s Theorem Embedding idempotent commutative latin squares: Cruse’s Theorem
- 9.5 Embedding partial Steiner triple systems
- 10 Steiler Quadruple Systems
- 10.1 Introduction
- 10.2 Constructions of Steiner Quadruple Systems
- 10.3 The Stern and Lenz Lemma
- 10.4 The (3v - 2u)‐Construction)‐Construction
- Appendices
- Appendices A: Cyclic Steiner Triple Systems
- Appendices B: Answers to Selected Exercises
- References