Erdõs–Ko–Rado Theorems: Algebraic Approaches

Graduate text focusing on algebraic methods that can be applied to prove the Erd?s–Ko–Rado Theorem and its generalizations.

Christopher Godsil (Author), Karen Meagher (Author)

9781107128446, Cambridge University Press

Hardback, published 24 November 2015

350 pages, 5 b/w illus. 170 exercises
23.5 x 15.8 x 2.3 cm, 0.62 kg

'This is an excellent book about Erdos-Ko-Rado (EKR) Theorems and how to prove them by algebraic methods … The writing style is reader-friendly, and proofs are well organized and easily followed. Also, every chapter contains Exercises and Notes, which are very useful for expanding understanding and finding further reading. The reviewer recommends this book without hesitation to all graduate students and researchers interested in combinatorics.' Norihide Tokushige, MathSciNet

Aimed at graduate students and researchers, this fascinating text provides a comprehensive study of the Erd?s–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.

Preface
1. The Erd?s–Ko–Rado 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.

Subject Areas: Combinatorics & graph theory [PBV], Algebra [PBF], Discrete mathematics [PBD], Mathematics [PB]