Extremal Combinatorics

Introduction.- I. The Classis: Counting.- The Pigeon-Hole Principle.- Principle of Inclusion and Exclusion.- Systems of Distinct Representatives.- Colorings.- Chains and Antichains.- Intersecting Families.- Covers and Transversals.- Sunflowers.- Density and Universality.- Designs.- Witness Sets.- Isolation Lemmas.- II. The Linear Algebra Method: Basic Method.- The Polynomial Technique.- Monotone Span Programs.- III. The Probabilistic Method: Basic Tools.- Counting Sieve.- Lovasz Sieve.- Linearity of Expectation.- The Deletion Method.- Second Moment Method.- Bounding of Large Deviations.- Randomized Algorithms.- Derandomization.- The Entropy Function.- Random Walks and Search Problems.- IV. Fragments of Ramsey Theory: Ramsey's Theorem.- The Hales-Jewett Theorem.- Epilogue: What Next?- Bibliography.- Index.

From the reviews:"This monograph deals with problems of the following type: 'If a collection of finite objects ... satisfies certain restrictions, how large or how small can it be?' The text is self-contained and assumes no special knowledge (only a standard mathematical background). Moreover, its 29 chapters - 'each devoted to a particular proof technique' - are (almost) independent. Because of this modularity, its style ... and the character of its subject, this book can also be browsed and read for (mathematical) pleasure." (P. Schmitt, Monatshefte für Mathematik, Vol. 141 (1), 2004)"The book is structured as a collection of short, largely independent chapters, each dedicated to a specific proof technique. ... The book is broad in scope and gives equal space to the classical counting techniques and to more recent methods. ... I used the book to teach a small group of graduate students. It was a rewarding experience. ... The material was interesting, diverse, and challenging. ... this is a book I heartily recommend to anyone wishing to learn or teach combinatorics." (Jeannette C. M. Janssen, SIAM Review, Vol. 26 (1), 2004)"The author has covered a huge amount of ground in this book. ... Each topic is covered in a way that takes the reader from the start right up to the most recent results in the area. ... the book is clearly a 'labour of love'. The author's enthusiasm for the subject shines through page after page: it is hard not to feel his excitement as one reads what he has written." (Imre Leader, Combinatorics, Probability and Computing, Vol. 13, 2004)"A text suitable for advanced undergraduate or graduate students. It begins with basics, inclusion-exclusion, pigeonhole, systems of distinct representatives. Substantial space is devoted to the more modern linear algebra and probabilistic methods. Algorithmic aspects permeate the book, making it suitable for a computer science, or joint math/computer science course. There are numerous exercises." (J.