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.
Arvustused
'A treasure trove of material for teachers of advanced courses on extremal graphs and hypergraphs. The book gives a thoughtfully curated presentation of this vibrant field, from its classical beginnings right through to present-day research developments. It is sure to become an essential resource for students, researchers and educators.' Penny Haxell, University of Waterloo 'Extremal Graph and Hypergraph Theory is a vibrant area of modern Discrete Mathematics. Mubayi and Verstraete describe the major themes of the subject, and provide a beautiful presentation of many of the most exciting recent developments. Written by two leading experts, this fascinating text will benefit students and researchers interested in the field and its applications.' Noga Alon, Princeton University
Muu info
A cohesive reference covering modern developments and breakthroughs in extremal graph, hypergraph and Ramsey theory.
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.
Dhruv Mubayi is Professor in the Department of Mathematics, Statistics, and Computer Science at the University of Illinois Chicago. He has published over 150 research papers in combinatorics, received research fellowships from the Sloan, Simons, and Humboldt foundations and is a Fellow of the American Mathematical Society. Jacques A. Verstraete is Professor of Mathematics at the University of California, San Diego. His research is in extremal combinatorics and Ramsey theory. He is a recipient of awards including an Alfred Sloan Research Fellowship and Frontiers of Science Award.
