This book is a concise, self-contained, up-to-date introduction to extremal combinatorics for nonspecialists. There is a strong emphasis on theorems with particularly elegant and informative proofs, they may be called gems of the theory. The author presents a wide spectrum of the most powerful combinatorial tools together with impressive applications in computer science: methods of extremal set theory, the linear algebra method, the probabilistic method, and fragments of Ramsey theory. No special knowledge in combinatorics or computer science is assumed – the text is self-contained and the proofs can be enjoyed by undergraduate students in mathematics and computer science. Over 300 exercises of varying difficulty, and hints to their solution, complete the text.
This second edition has been extended with substantial new material, and has been revised and updated throughout. It offers three new chapters on expander graphs and eigenvalues, the polynomial method and error-correcting codes. Most of the remaining chapters also include new material, such as the Kruskal—Katona theorem on shadows, the Lovász—Stein theorem on coverings, large cliques in dense graphs without induced 4-cycles, a new lower bounds argument for monotone formulas, Dvir's solution of the finite field Kakeya conjecture, Moser's algorithmic version of the Lovász Local Lemma, Schöning's algorithm for 3-SAT, the Szemerédi—Trotter theorem on the number of point-line incidences, surprising applications of expander graphs in extremal number theory, and some other new results.
Texts in Theoretical Computer Science An EATCS Series Editors: J. Hromkoviˇc G. Rozenberg A. Salomaa Founding Editors: W. Brauer G. Rozenberg A. Salomaa On behalf of the European Association for Theoretical Computer Science (EATCS)
Advisory Board: G. Ausiello M. Broy C.S. Calude A. Condon D. Harel J. Hartmanis T. Henzinger T. Leighton M. Nivat C. Papadimitriou D. Scott
For further volumes: www.springer.com/series/3214
Stasys Jukna
Extremal Combinatorics With Applications in Computer Science Second Edition
Prof. Dr. Stasys Jukna Goethe Universität Frankfurt Institut für Informatik Robert-Mayer Str. 11-15 60054 Frankfurt am Main Germany
[email protected] Vilnius University Institute of Mathematics and Informatics Akademijos 4 08663 Vilnius Lithuania Series Editors Prof. Dr. Juraj Hromkoviˇc ETH Zentrum Department of Computer Science Swiss Federal Institute of Technology 8092 Zürich, Switzerland
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Prof. Dr. Grzegorz Rozenberg Leiden Institute of Advanced Computer Science University of Leiden Niels Bohrweg 1 2333 CA Leiden, The Netherlands
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Prof. Dr. Arto Salomaa Turku Centre of Computer Science Lemminkäisenkatu 14 A 20520 Turku, Finland
[email protected]fi
ISSN 1862-4499 Texts in Theoretical Computer Science. An EATCS Series ISBN 978-3-642-17363-9 e-ISBN 978-3-642-17364-6 DOI 10.1007/978-3-642-17364-6 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011937551 ACM Codes: G.2, G.3, F.1, F.2, F.4.1 © Springer-Verlag Berlin Heidelberg 2001, 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse