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The book is a concise, self-contained and up-to-date introduction to extremal combinatorics for non-specialists. Strong emphasis is made on theorems with particularly elegant and informative proofs which may be called gems of the theory. A wide spectrum of most powerful combinatorial tools is presented: methods of extremal set theory, the linear algebra method, the probabilistic method and fragments of Ramsey theory. A throughout discussion of some recent applications to computer science motivates the liveliness and inherent usefulness of these methods to approach problems outside combinatorics. No special combinatorial or algebraic background is assumed. All necessary elements of linear algebra and discrete probability are introduced before their combinatorial applications. Aimed primarilrily as an introductory text for graduates, it provides also a compact source of modern extremal combinatorics for researchers in computer science and other fields of discrete mathematics.
List of contents
Contents:
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
(Lov..sz) 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
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