Extremal Combinatorics

1. Front Matter

   Pages I-XVII

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2. Prolog: What This Book Is About

   1. Prolog: What This Book Is About

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      Pages 1-4

3. Notation

   1. Notation

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      Pages 5-8

4. The Classics

   1. Front Matter

      Pages 9-9

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   2. Counting

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      Pages 11-22

   3. Advanced Counting

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      Pages 23-31

   4. The Principle of Inclusion and Exclusion

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      Pages 32-36

   5. The Pigeonhole Principle

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      Pages 37-54

   6. Systems of Distinct Representatives

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      Pages 55-64

   7. Colorings

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      Pages 65-76

5. Extremal Set Theory

   1. Front Matter

      Pages 77-77

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   2. Sunflowers

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      Pages 79-88

   3. Intersecting Families

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      Pages 89-96

   4. Chains and Antichains

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      Pages 97-108

   5. Blocking Sets and the Duality

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      Pages 109-132

   6. Density and Universality

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      Pages 133-142

   7. Witness Sets and Isolation

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      Pages 143-152

   8. Designs

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      Pages 153-166

6. The Linear Algebra Method

   1. Front Matter

      Pages 167-167

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   2. The Basic Method

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      Pages 169-190

Combinatorial mathematics has been pursued since time immemorial, and at a reasonable scientific level at least since Leonhard Euler (1707-1783). It ren­ dered many services to both pure and applied mathematics. Then along came the prince of computer science with its many mathematical problems and needs - and it was combinatorics that best fitted the glass slipper held out. Moreover, it has been gradually more and more realized that combinatorics has all sorts of deep connections with "mainstream areas" of mathematics, such as algebra, geometry and probability. This is why combinatorics is now apart of the standard mathematics and computer science curriculum. This book is as an introduction to extremal combinatorics - a field of com­ binatorial mathematics which has undergone aperiod of spectacular growth in recent decades. The word "extremal" comes from the nature of problems this field deals with: if a collection of finite objects (numbers, graphs, vectors, sets, etc. ) satisfies certain restrictions, how large or how small can it be? For example, how many people can we invite to a party where among each three people there are two who know each other and two who don't know each other? An easy Ramsey-type argument shows that at most five persons can attend such a party. Or, suppose we are given a finite set of nonzero integers, and are asked to mark an as large as possible subset of them under the restriction that the sum of any two marked integers cannot be marked.

• Diskrete Mathematik
• Kombinatorik
• algorithms
• combinatorics
• computational complexity
• discrete mathematics
• extremal combinatorics
• extremal set theory
• linear algebra

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 jointmath/computer science course. There are numerous exercises." (J. Spencer, Mathematical Reviews, Issue 2003 g)

"Extremal Combinatorics is a part of finite mathematics … . The present book collects many different aspects of the field. It is wider than deep having 29 relatively short and independent chapters. These properties make the book accessible to a broad readership. … this volume also contains a large number of well chosen exercises of various range of difficulty. There is a useful home page edited by the author … . We warmly recommend this well-written and nicely edited book to anybody with combinatorial interest." (János Barát, Acta Scientiarum Mathematicarum, Vol. 68, 2002)

"This book presents several important parts of combinatorics with emphasis to methods for solving extremal problems. Some interesting applications in theoretical computer science are included. … As written in the preface, the text is indeed self-contained and the chapters are almost independent. More than 300 exercises … are included. The presentation is clear and sound. The book is not only valuable for students and teachers, but also for researchers working in discrete mathematics or theoretical computer science." (Konrad Engel, Zentralblatt MATH, Vol. 978, 2002)

• Informatik - FB 20, Theoretische Informatik, Johann Wolfgang Goethe-Universität, Frankfurt, Germany

  Stasys Jukna

• Institute of Mathematics and Informatics, Vilnius, Lethuania

  Stasys Jukna

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