Ramsey Theory for Discrete Structures

Author: Hans Jürgen Prömel
Publisher: Springer Science & Business Media
ISBN: 9783319013152
Release Date: 2013-12-04
Genre: Mathematics

This monograph covers some of the most important developments in Ramsey theory from its beginnings in the early 20th century via its many breakthroughs to recent important developments in the early 21st century. The book first presents a detailed discussion of the roots of Ramsey theory before offering a thorough discussion of the role of parameter sets. It presents several examples of structures that can be interpreted in terms of parameter sets and features the most fundamental Ramsey-type results for parameter sets: Hales-Jewett's theorem and Graham-Rothschild1s Ramsey theorem as well as their canonical versions and several applications. Next, the book steps back to the most basic structure, to sets. It reviews classic results as well as recent progress on Ramsey numbers and the asymptotic behavior of classical Ramsey functions. In addition, it presents product versions of Ramsey's theorem, a combinatorial proof of the incompleteness of Peano arithmetic, provides a digression to discrepancy theory and examines extensions of Ramsey's theorem to larger cardinals. The next part of the book features an in-depth treatment of the Ramsey problem for graphs and hypergraphs. It gives an account on the existence of sparse and restricted Ramsey theorem's using sophisticated constructions as well as probabilistic methods. Among others it contains a proof of the induced Graham-Rothschild theorem and the random Ramsey theorem. The book closes with a chapter on one of the recent highlights of Ramsey theory: a combinatorial proof of the density Hales-Jewett theorem. This book provides graduate students as well as advanced researchers with a solid introduction and reference to the field.

Ramsey Theory

Author: Ronald L. Graham
Publisher: John Wiley & Sons
ISBN: 0471500461
Release Date: 1990-03-16
Genre: Mathematics

"Discrete mathematics, the study of finite structures, is one of the fastest-growing areas in mathematics. The wide applicability of its evolving techniques points to the rapidity with which the field is moving from its beginnings to its maturity, and reflects the ever-increasing interaction between discrete mathematics and computer science. This Series provides broad coverage of discrete mathematics and optimization, ranging over such fields as combinatorics, graph theory, enumeration, and the analysis of algorithms." -- Book cover.

Mathematics of Ramsey theory

Author: Jaroslav Nes̆etřil
Publisher: Springer Verlag
ISBN: UCAL:B5008670
Release Date: 1990
Genre: Mathematics

One of the important areas of contemporary combinatorics is Ramsey theory. Ramsey theory is basically the study of structure preserved under partitions. The general philosophy is reflected by its interdisciplinary character. The ideas of Ramsey theory are shared by logicians, set theorists and combinatorists, and have been successfully applied in other branches of mathematics. The whole subject is quickly developing and has some new and unexpected applications in areas as remote as functional analysis and theoretical computer science. This book is a homogeneous collection of research and survey articles by leading specialists. It surveys recent activity in this diverse subject and brings the reader up to the boundary of present knowledge. It covers virtually all main approaches to the subject and suggests various problems for individual research.

Discrete Mathematics with Proof

Author: Eric Gossett
Publisher: John Wiley & Sons
ISBN: 9780470457931
Release Date: 2009-06-22
Genre: Mathematics

"Discrete mathematics has become increasingly popular in recent years due to its growing applications in the field of computer science. - Discrete Mathematics with Proof, Second Edition continues to facilitate an up-to-date understanding of this important topic, exposing readers to a wide range of modern and technological applications. Assuming only a basic background in calculus, Discrete Mathematics with Proof, Second Edition is an excellent book for mathematics and computer science courses at the undergraduate level. - It is also a valuable resource for professionals in various technical fields who would like an introduction to discrete mathematics."--Jacket.

Contemporary Trends in Discrete Mathematics

Author: Ronald L. Graham
Publisher: American Mathematical Soc.
ISBN: 0821885812
Release Date: 1999-01-01
Genre: Mathematics

Discrete mathematics stands among the leading disciplines of mathematics and theoretical computer science. This is due primarily to its increasing role in university curriculae and its growing importance in applications ranging from optimization to molecular biology. An inaugural conference was held cooperatively by DIMATIA and DIMACS to focus on the versatility, width, and depth of current progress in the subject area. This volume offers a well-balanced blend of research and survey papers reflecting the exciting, attractive topics in contemporary discrete mathematics. Discussed in the book are topics such as graph theory, partially ordered sets, geometrical Ramsey theory, computational complexity issues and applications.

Ramsey Theory

Author: Alexander Soifer
Publisher: Springer Science & Business Media
ISBN: 0817680926
Release Date: 2010-10-29
Genre: Mathematics

This book explores the theory’s history, recent developments, and some promising future directions through invited surveys written by prominent researchers in the field. The first three surveys provide historical background on the subject; the last three address Euclidean Ramsey theory and related coloring problems. In addition, open problems posed throughout the volume and in the concluding open problem chapter will appeal to graduate students and mathematicians alike.

Mathematics of Ramsey Theory

Author: Jaroslav Nesetril
Publisher: Springer Science & Business Media
ISBN: 9783642729058
Release Date: 2012-12-06
Genre: Mathematics

One of the important areas of contemporary combinatorics is Ramsey theory. Ramsey theory is basically the study of structure preserved under partitions. The general philosophy is reflected by its interdisciplinary character. The ideas of Ramsey theory are shared by logicians, set theorists and combinatorists, and have been successfully applied in other branches of mathematics. The whole subject is quickly developing and has some new and unexpected applications in areas as remote as functional analysis and theoretical computer science. This book is a homogeneous collection of research and survey articles by leading specialists. It surveys recent activity in this diverse subject and brings the reader up to the boundary of present knowledge. It covers virtually all main approaches to the subject and suggests various problems for individual research.

Connections in Discrete Mathematics

Author: Steve Butler
Publisher: Cambridge University Press
ISBN: 9781107153981
Release Date: 2018-05-31
Genre: Mathematics

Many of the best researchers and writers in discrete mathematics come together in a volume inspired by Ron Graham.

Einf hrung in die Geometrie und Topologie

Author: Werner Ballmann
Publisher: Springer-Verlag
ISBN: 9783034809016
Release Date: 2015-02-19
Genre: Mathematics

Das Buch bietet eine Einführung in die Topologie, Differentialtopologie und Differentialgeometrie. Es basiert auf Manuskripten, die in verschiedenen Vorlesungszyklen erprobt wurden. Im ersten Kapitel werden grundlegende Begriffe und Resultate aus der mengentheoretischen Topologie bereitgestellt. Eine Ausnahme hiervon bildet der Jordansche Kurvensatz, der für Polygonzüge bewiesen wird und eine erste Idee davon vermitteln soll, welcher Art tiefere topologische Probleme sind. Im zweiten Kapitel werden Mannigfaltigkeiten und Liesche Gruppen eingeführt und an einer Reihe von Beispielen veranschaulicht. Diskutiert werden auch Tangential- und Vektorraumbündel, Differentiale, Vektorfelder und Liesche Klammern von Vektorfeldern. Weiter vertieft wird diese Diskussion im dritten Kapitel, in dem die de Rhamsche Kohomologie und das orientierte Integral eingeführt und der Brouwersche Fixpunktsatz, der Jordan-Brouwersche Zerlegungssatz und die Integralformel von Stokes bewiesen werden. Das abschließende vierte Kapitel ist den Grundlagen der Differentialgeometrie gewidmet. Entlang der Entwicklungslinien, die die Geometrie der Kurven und Untermannigfaltigkeiten in Euklidischen Räumen durchlaufen hat, werden Zusammenhänge und Krümmung, die zentralen Konzepte der Differentialgeometrie, diskutiert. Den Höhepunkt bilden die Gaussgleichungen, die Version des theorema egregium von Gauss für Untermannigfaltigkeiten beliebiger Dimension und Kodimension. Das Buch richtet sich in erster Linie an Mathematik- und Physikstudenten im zweiten und dritten Studienjahr und ist als Vorlage für ein- oder zweisemestrige Vorlesungen geeignet.

Local search in combinatorial optimization

Author: J. K. Lenstra
Publisher: John Wiley & Sons
ISBN: 0471948225
Release Date: 1997-06-17
Genre: Mathematics

Wiley-Interscience Series in Discrete Mathematics and Optimization Advisory Editors Ronald L. Graham Jan Karel Lenstra Robert E. Tarjan Discrete Mathematics and Optimization involves the study of finite structures and is one of the fastest growing areas in mathematics today. The level and depth of recent advances in the area and the wide applicability of its evolving techniques point to the rapidity with which the field is moving and presage the ever-increasing interaction between it and computer science. The Series provides a broad coverage of discrete mathematics and optimization, ranging over such fields as combinatorics, graph theory, enumeration, mathematical programming and the analysis of algorithms, and including such topics as Ramsey theory, transversal theory, block designs, finite geometries, Polya theory, graph and matroid algorithms, network flows, polyhedral combinatorics and computational complexity. The Wiley-Interscience Series in Discrete Mathematics and Optimization will be a substantial part of the record in this extraordinary development. Recent title in the Series: Theory and Algorithms for Linear Optimization: An Interior Point Approach C. Roos, T. Terlaky Delft University of Technology, The Netherlands and J.-Ph. Vial University of Geneva, Switzerland Linear Optimization (LO) is one of the most widely taught and fast developing techniques in mathematics, with applications in many areas of science, commerce and industry. The dramatically increased interest in the subject is due mainly to advances in computer technology and to the development of Interior Point Methods (IPM) for LO. This book provides a unified presentation of the field by way of an interior point approach to both the theory of LO and algorithms for LO (design, covergence, complexity and asymptotic behaviour). A common thread throughout the book is the role of strictly complementary solutions, which play a crucial role in the interior point approach and distinguishes the new approach from the classical Simplex-based approach. The approach to LO in this book is new in many aspects. In particular the IPM based development of duality theory is surprisingly elegant. The algorithmic parts of the book contain a complete discussion of many algorithmic variants, including predictor-corrector methods, partial updating, higher order methods and sensitivity and parametric analysis. The comprehensive and up-to-date coverage of the subject, together with the clarity of presentation, ensures that this book will be an invaluable resource for researchers and professionals who wish to develop their understanding of LOs and IPMs . Numerous exercises are provided to help consolidate understanding of the material and more than 45 figures are included to illustrate the characteristics of the algorithms. A general understanding of linear algebra and calculus is assumed and the preliminary chapters provide a self-contained introduction for readers who are unfamiliar with LO methods. These chapters will also be of interest for readers who wish to take a fresh look at the topics. 1997

Discrete Mathematics

Author: CTI Reviews
Publisher: Cram101 Textbook Reviews
ISBN: 9781467293969
Release Date: 2016-10-16
Genre: Education

Facts101 is your complete guide to Discrete Mathematics. In this book, you will learn topics such as as those in your book plus much more. With key features such as key terms, people and places, Facts101 gives you all the information you need to prepare for your next exam. Our practice tests are specific to the textbook and we have designed tools to make the most of your limited study time.

A Beginner s Guide to Graph Theory

Author: W.D. Wallis
Publisher: Springer Science & Business Media
ISBN: 0817644849
Release Date: 2007-06-08
Genre: Mathematics

Concisely written, gentle introduction to graph theory suitable as a textbook or for self-study Graph-theoretic applications from diverse fields (computer science, engineering, chemistry, management science) 2nd ed. includes new chapters on labeling and communications networks and small worlds, as well as expanded beginner's material Many additional changes, improvements, and corrections resulting from classroom use

Pearls of Discrete Mathematics

Author: Martin Erickson
Publisher: CRC Press
ISBN: 9781439816172
Release Date: 2009-09-16
Genre: Computers

Methods Used to Solve Discrete Math Problems Interesting examples highlight the interdisciplinary nature of this area Pearls of Discrete Mathematics presents methods for solving counting problems and other types of problems that involve discrete structures. Through intriguing examples, problems, theorems, and proofs, the book illustrates the relationship of these structures to algebra, geometry, number theory, and combinatorics. Each chapter begins with a mathematical teaser to engage readers and includes a particularly surprising, stunning, elegant, or unusual result. The author covers the upward extension of Pascal’s triangle, a recurrence relation for powers of Fibonacci numbers, ways to make change for a million dollars, integer triangles, the period of Alcuin’s sequence, and Rook and Queen paths and the equivalent Nim and Wythoff’s Nim games. He also examines the probability of a perfect bridge hand, random tournaments, a Fibonacci-like sequence of composite numbers, Shannon’s theorems of information theory, higher-dimensional tic-tac-toe, animal achievement and avoidance games, and an algorithm for solving Sudoku puzzles and polycube packing problems. Exercises ranging from easy to challenging are found in each chapter while hints and solutions are provided in an appendix. With over twenty-five years of teaching experience, the author takes an organic approach that explores concrete problems, introduces theory, and adds generalizations as needed. He delivers an absorbing treatment of the basic principles of discrete mathematics.

Das BUCH der Beweise

Author: Martin Aigner
Publisher: Springer-Verlag
ISBN: 9783662577677
Release Date: 2018-09-17
Genre: Mathematics

Diese fünfte deutsche Auflage enthält ein ganz neues Kapitel über van der Waerdens Permanenten-Vermutung, sowie weitere neue, originelle und elegante Beweise in anderen Kapiteln. Aus den Rezensionen: “... es ist fast unmöglich, ein Mathematikbuch zu schreiben, das von jedermann gelesen und genossen werden kann, aber Aigner und Ziegler gelingt diese Meisterleistung in virtuosem Stil. [...] Dieses Buch erweist der Mathematik einen unschätzbaren Dienst, indem es Nicht-Mathematikern vorführt, was Mathematiker meinen, wenn sie über Schönheit sprechen.” Aus der Laudatio für den “Steele Prize for Mathematical Exposition” 2018 "Was hier vorliegt ist eine Sammlung von Beweisen, die in das von Paul Erdös immer wieder zitierte BUCH gehören, das vom lieben (?) Gott verwahrt wird und das die perfekten Beweise aller mathematischen Sätze enthält. Manchmal lässt der Herrgott auch einige von uns Sterblichen in das BUCH blicken, und die so resultierenden Geistesblitze erhellen den Mathematikeralltag mit eleganten Argumenten, überraschenden Zusammenhängen und unerwarteten Volten." www.mathematik.de, Mai 2002 "Eine einzigartige Sammlung eleganter mathematischer Beweise nach der Idee von Paul Erdös, verständlich geschrieben von exzellenten Mathematikern. Dieses Buch gibt anregende Lösungen mit Aha-Effekt, auch für Nicht-Mathematiker." www.vismath.de "Ein prächtiges, äußerst sorgfältig und liebevoll gestaltetes Buch! Erdös hatte die Idee DES BUCHES, in dem Gott die perfekten Beweise mathematischer Sätze eingeschrieben hat. Das hier gedruckte Buch will eine "very modest approximation" an dieses BUCH sein.... Das Buch von Aigner und Ziegler ist gelungen ..." Mathematische Semesterberichte, November 1999 "Wer (wie ich) bislang vergeblich versucht hat, einen Blick ins BUCH zu werfen, wird begierig in Aigners und Zieglers BUCH der Beweise schmökern." www.mathematik.de, Mai 2002