Erd¿s-Ko-Rado Theorems

Christopher Godsil is a professor in the Combinatorics and Optimization Department at the University of Waterloo, Ontario. He authored (with Gordon Royle) the popular textbook Algebraic Graph Theory. He started the Journal of Algebraic Combinatorics in 1992 and he serves on the editorial board of a number of other journals, including the Australasian Journal of Combinatorics and the Electronic Journal of Combinatorics.

Aimed at graduate students and researchers, this fascinating text provides a comprehensive study of the Erdos-Ko-Rado Theorem, with a focus on algebraic methods. The authors begin by discussing well-known proofs of the EKR bound for intersecting families. The natural generalization of the EKR Theorem holds for many different objects that have a notion of intersection, and the bulk of this book focuses on algebraic proofs that can be applied to these different objects. The authors introduce tools commonly used in algebraic graph theory and show how these can be used to prove versions of the EKR Theorem. Topics include association schemes, strongly regular graphs, the Johnson scheme, the Hamming scheme and the Grassmann scheme. Readers can expand their understanding at every step with the 170 end-of-chapter exercises. The final chapter discusses in detail 15 open problems, each of which would make an interesting research project.

The Erdos-Ko-Rado Theorem is a fundamental result in combinatorics. Aimed at graduate students and researchers, this comprehensive text shows how tools from algebraic graph theory can be applied to prove the EKR Theorem and its generalizations. Readers can test their understanding at every step with the end-of-chapter exercises.

Preface; 1. The Erd¿s¿KöRado Theorem; 2. Bounds on cocliques; 3. Association schemes; 4. Distance-regular graphs; 5. Strongly regular graphs; 6. The Johnson scheme; 7. Polytopes; 8. The exact bound; 9. The Grassmann scheme; 10. The Hamming scheme; 11. Representation theory; 12. Representations of symmetric group; 13. Orbitals; 14. Permutations; 15. Partitions; 16. Open problems; Glossary of symbols; Glossary of operations and relations; References; Index.