Steiner Triple Systems
The Existence Problem
v ≡ 3 (mod 6): The Bose Construction
v ≡ 1 (mod 6): The Skolem Construction
v ≡ 5 (mod 6): The 6n + 5 Construction
Quasigroups with Holes and Steiner Triple Systems
The Wilson Construction
Cyclic Steiner Triple Systems
The 2n + 1 and 2n + 7 Constructions
λ-Fold Triple Systems
Triple Systems of Index λ > 1
The Existence of Indempotent Latin Squares
2-fold Triple Systems
λ= 3 and 6
λ-Fold Triple Systems in General
Quasigroup Identities and Graph Decompositions
Quasigroup Identities
Mendelsohn Triple Systems Revisited
Steiner Triple Systems Revisited
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 ≡ 2 (mod 4)
Affine and Projective Planes
Affine Planes
Projective Planes
Connections between Affine and Projective Planes
Connection between Affine Planes and Complete Sets of MOLS
Coordinating the Affine Plane
Intersections of Steiner Triple Systems
Teirlinck’s Algorithm
The General Intersection Problem
Embeddings
Embedding Latin Rectangles—Necessary Conditions
Edge-Coloring Bipartite Graphs
Embedding Latin Rectangles: Ryser’s Sufficient Conditions
Embedding Idempotent Commutative Latin Squares: Cruse’s Theorem
Embedding Partial Steiner Triple Systems
Steiner Quadruple Systems
Introduction
Constructions of Steiner Quadruple Systems
The Stern and Lenz Lemma
The (3v – 2u)-Construction
Appendix A: Cyclic Steiner Triple Systems
Appendix B: Answers to Selected Exercises
References
Index
Biography
Charles C. Lindner, Christopher A. Rodger
…it is remarkable how quickly the book propels the reader from the basics to the frontiers of design theory … Combined, these features make the book an excellent candidate for a design theory text. At the same time, even the seasoned researcher of triple systems will find this a useful resource.
—Peter James Dukes (3-VCTR-MS; Victoria, BC), Mathematical Reviews, 2010
