Design Theory is not an official ETSU class. It is modeled on the Auburn University undergraduate/graduate class "Combinatorial Designs" (MATH 5770/6770). The 2021-22 Auburn Bulletin description of Combinatorial Designs is: "Latin squares, mutually orthogonal latin squares, orthogonal and perpendicular arrays, Steiner triple systems, block designs, difference sets and finite geometries." The prerequisite should be ETSU's Mathematical Reasoning (MATH 3000) for the introduction to proof techniques and the exposure to introductory counting techniques and elementary number theory. Some exposure to combinatorial ideas, such as those covered in Applied Combinatorics and Problem Solving (MATH 3340), and to graph theoretic ideas, such as those given in Introduction to Graph Theory (MATH 4347/5347), would be helpful. Notice that the authors of the text book, Drs. Curt C. Lindner (July 21, 1938-February 21, 2023) and Chris A. Rodger, are both emeritis professors from Auburn University's Department of Mathematics and Statistics. I was a graduate student in Auburn University's (sadly no longer existent) "Department of Algebra, Combinatorics, and Analysis" and finished my master's degree with a thesis (available online) titled Automorphisms of Steiner Triple Systems. Though I never took a class from either of the authors, Chris Rodger was on my master's thesis committee and Curt Lindner attended my master's thesis defense. By the way, much of this material was covered in a class I took at Auburn University in fall 1985, Combinatorial Theory 1 (MH 673), taught by my M.S. thesis advisor, the inimitable Dean Hoffman (May 8, 1949-November 11, 2022).

Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).

• Preface.
• Chapter 1. Steiner Triple Systems.
• Chapter 2. λ-Fold Triple Systems.
• Chapter 3. Quasigroup Identities and Graph Decompositions.
• Chapter 4. Maximum Packings and Minimum Coverings.
• Chapter 5. Kirkman Triple Systems.
• Chapter 6. Mutually Orthogonal Latin Squares.
• Chapter 7. Affine and Projective Planes.
• Chapter 8. Intersections of Steiner Triple Systems.
• Chapter 9. Embeddings.
• Chapter 10. Steiner Quadruple Systems.
• Appendices.

Preface. PDF

Chapter 1. Steiner Triple Systems.

• Section 1.1. The Existence Problem. Section 1.1 notes
• Supplement. Proofs from Section 1.1. Proofs of Theorems in Section 1.1
• Supplement. Printouts of Proofs from Section 1.1. Print file for Proofs of Theorems in Section 1.1
• Section 1.2. v ≡ 3 (mod 6): The Bose Construction. Section 1.2 notes
• Supplement. Proofs from Section 1.2. Proofs of Theorems in Section 1.2
• Supplement. Printouts of Proofs from Section 1.2. Print file for Proofs of Theorems in Section 1.2
• Section 1.3. v ≡ 1 (mod 6): The Skolem Construction. Section 1.3 notes
• Section 1.4. v ≡ 5 (mod 6): The 6n + 5 Construction. Section 1.4 notes (This section introduces pairwise balanced designs.)
• Section 1.5. Quasigroups with Holes and Steiner Triple Systems. Section 1.5 notes
• Supplement. Proofs from Section 1.5. Proofs of Theorems in Section 1.5
• Supplement. Printouts of Proofs from Section 1.5. Print file for Proofs of Theorems in Section 1.5
• Section 1.6. The Wilson Construction. Section 1.6 notes
• Supplement. Proofs from Section 1.6. Proofs of Theorems in Section 1.6
• Supplement. Printouts of Proofs from Section 1.6. PDF
• Section 1.7. Cyclic Steiner Triple Systems. Section 1.7 notes
• Supplement. Proofs from Section 1.7. Proofs of Theorems in Section 1.7
• Supplement. Printouts of Proofs from Section 1.7. Print file for Proofs of Theorems in Section 1.7
• Section 1.8. The 2n + 1 and 2n + 7 Constructions (partial). Section 1.8 notes
• Study Guide 1

Chapter 2. λ-Fold Triple Systems.

• Section 2.1. Triple Systems of Index λ > 1. Section 2.1 notes
• Section 2.2. The Existence of Idempotent Latin Squares. Section 2.2 notes
• Supplement. Proofs from Section 2.2. Proofs of Theorems in Section 2.2
• Supplement. Printouts of Proofs from Section 2.2. Print file for Proofs of Theorems in Section 2.2
• Section 2.3. 2-Fold Triple Systems. Section 2.3 notes
• Supplement. Proofs from Section 2.3. Proofs of Theorems in Section 2.3
• Supplement. Printouts of Proofs from Section 2.3. Print file for Proofs of Theorems in Section 2.3
• Section 2.4. Mendelsohn Triple Systems. Section 2.4 notes
• Supplement. Directed and Hybrid Triple Systems. Direct and Hybrid Triple Systems notes
• Supplement. Mixed Triple Systems. Mixed Triple Systems notes (This section includes several unsolved problems concerning mixed graphs.)
• Section 2.5. λ = 3 and 6. Section 2.5 notes
• Section 2.6. λ-Fold Triple Systems in General. Section 2.6 notes
• Study Guide 2

Chapter 3. Quasigroup Identities and Graph Decompositions.

• Section 3.1. Quasigroup Identities. Section 3.1 notes
• Supplement. Proofs from Section 3.1. Proofs of Theorems in Section 3.1
• Supplement. Printouts of Proofs from Section 3.1. Print file for Proofs of Theorems in Section 3.1
• Section 3.2. Mendelsohn Triple Systems Revisited. Section 3.2 notes
• Section 3.3. Steiner Triple Systems Revisited. Section 3.3 notes
• Study Guide 3

Chapter 4. Maximum Packings and Minimum Coverings.

• Section 4.1. The General Problem. Section 4.1 notes
• Section 4.2. Maximum Packings. Section 4.2 notes
• Section 4.3. Minimum Coverings. Section 4.3 notes
• Supplement. Packings and Coverings for Mendelsohn and Directed Triple Systems. Packings and Coverings for MTS and DTS notes
• Supplement. Proofs from Supplement Packings and Coverings. Proofs of Theorems Packings and Coverings for MTS and DTS
• Supplement. Printouts of Proofs from Supplement Packings and Coverings. Print file for Proofs of Theorems Packings and Coverings for MTS and DTS
• Study Guide 4

Chapter 5. Kirkman Triple Systems.

• Section 5.1. A Recursive Construction.
• Section 5.2. Constructing Pairwise Balanced Designs.
• Study Guide 5.

Chapter 6. Mutually Orthogonal Latin Squares.

• Section 6.1. Introduction. Section 6.1 notes
• Supplement. Proofs from Section 6.1. Proofs of Theorems in Section 6.1
• Supplement. Printouts of Proofs from Section 6.1. Print file for Proofs of Theorems in Section 6.1
• Section 6.2. The Euler and MacNeish Conjectures.
• Section 6.3. Disproof of the MacNeish Conjecture.
• Section 6.4. Disproof of the Euler Conjecture.
• Section 6.5. Orthogonal Latin Squares of order n ≡ 2 (mod 4).
• Study Guide 6.

Chapter 7. Affine and Projective Planes.

• Section 7.1. Affine Planes. Section 7.1 notes
• Section 7.2. Projective Planes. Section 7.2 notes
• Section 7.3. Connections between Affine and Projective Planes. Section 7.3 notes
• Section 7.4. Connections between Affine Planes and Complete sets of MOLS. Section 7.4 notes
• Section 7.5. Coordinatizing the Affine Plane. Section 7.5 notes
• Study Guide 7

Chapter 8. Intersections of Steiner Triple Systems.

• Section 8.1. Teirlinck's Algorithm. Section 8.1 notes
• Supplement. Proofs from Section 8.1. Proofs of Theorems in Section 8.1
• Supplement. Printouts of Proofs from Section 8.1. Print file for Proofs of Theorems in Section 8.1
• Section 8.2. The General Intersection Problem.
• Study Guide 8.

Chapter 9. Embeddings.

• Section 9.1. Embedding Latin Rectangles - Necessary Conditions.
• Section 9.2. Edge-Coloring Bipartite Graphs.
• Section 9.3. Embedding Latin Rectangles: Ryser's Sufficient Conditions.
• Section 9.4. Embedding Idempotent Commutative Latin Squares: Cruse's Theorem.
• Section 9.5. Embedding Partial Steiner Triple Systems.
• Study Guide 9.

Chapter 10. Steiner Quadruple Systems.

• Section 10.1. Introduction.
• Section 10.2. Constructions of Steiner Quadruple Systems.
• Section 10.3. The Stern and Lenz Lemma.
• Section 10.4. The (3v - 2u)-Construction.
• Study Guide 10.

Appendices.

• Appendix A. Cyclic Steiner Triple Systems. Cyclic Steiner Triple Systems notes
• Appendix B. Answers to Selected Exercises.

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