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Providing a cohesive reference for advanced undergraduates, graduate students and even experienced researchers, this text contains both introductory and advanced material in extremal graph theory, hypergraph theory and Ramsey theory. Along the way, the book includes many modern proof techniques in the field such as the probabilistic method and algebraic methods. Several recent breakthroughs are presented with complete proofs, for example, recent results on the sunflower problem, and off-diagonal and geometric Ramsey theory. It is perhaps unique in containing material on both hypergraph regularity and containers. Featuring an extensive list of exercises, the text is suitable as a teaching text for a variety of courses in extremal combinatorics. Each of the two parts can form the basis of separate courses, and the majority of sections are designed to match the length of a single lecture.
List of contents
Part I. Extremal Graph Theory: Introduction to Part I; 1. Matchings, paths and cycles; 2. Probabilistic methods and random graphs; 3. Turán's theorem; 4. Extremal problems for bipartite graphs; 5. Moore graphs and the even cycle theorem; 6. Random constructions; 7. Szemerédi's regularity lemma; 8. Pseudorandom graphs; 9. Graph Ramsey theory; 10. pseudorandom graph constructions in Ramsey theory; Part II. Extremal Set Theory and Hypergraph Theory: Introduction to Part II; 11. Posets; 12. Intersecting families; 13. Shadows and matchings; 14. Linear algebra method; 15. Algebraic methods; 16. Sunflowers; 17. The Turán problem for hypergraphs; 18. Matchings in linear hypergraphs; 19. Independent sets in hypergraphs; 20. Hypergraph regularity; 21. Hypergraph containers; 22. Hypergraph Ramsey theory; References; Glossary of notation; Index.
