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

Praise for the First Edition ". . . fills a considerable gap in the numerical analysis literature by providing a self-contained treatment . . . this is an important work written in a clear style . . . warmly recommended to any graduate student or researcher in the field of the numerical solution of partial differential equations." ?SIAM Review Time-Dependent Problems and Difference Methods, Second Edition continues to provide guidance for the analysis of difference methods for computing approximate solutions to partial differential equations for time-dependent problems. The book treats differential equations and difference methods with a parallel development, thus achieving a more useful analysis of numerical methods. The Second Edition presents hyperbolic equations in great detail as well as new coverage on second-order systems of wave equations including acoustic waves, elastic waves, and Einstein equations. Compared to first-order hyperbolic systems, initial-boundary value problems for such systems contain new properties that must be taken into account when analyzing stability. Featuring the latest material in partial differential equations with new theorems, examples, and illustrations,Time-Dependent Problems and Difference Methods, Second Edition also includes: High order methods on staggered grids Extended treatment of Summation By Parts operators and their application to second-order derivatives Simplified presentation of certain parts and proofs Time-Dependent Problems and Difference Methods, Second Edition is an ideal reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs and to predict and investigate physical phenomena. The book is also excellent for graduate-level courses in applied mathematics and scientific computations.

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.

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.

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 on the Integers

Author: Bruce M. Landman
Publisher: American Mathematical Soc.
ISBN: 9780821898673
Release Date: 2014-11-10
Genre: Mathematics

Ramsey theory is the study of the structure of mathematical objects that is preserved under partitions. In its full generality, Ramsey theory is quite powerful, but can quickly become complicated. By limiting the focus of this book to Ramsey theory applied to the set of integers, the authors have produced a gentle, but meaningful, introduction to an important and enticing branch of modern mathematics. Ramsey Theory on the Integers offers students a glimpse into the world of mathematical research and the opportunity for them to begin pondering unsolved problems. For this new edition, several sections have been added and others have been significantly updated. Among the newly introduced topics are: rainbow Ramsey theory, an "inequality" version of Schur's theorem, monochromatic solutions of recurrence relations, Ramsey results involving both sums and products, monochromatic sets avoiding certain differences, Ramsey properties for polynomial progressions, generalizations of the Erdős-Ginzberg-Ziv theorem, and the number of arithmetic progressions under arbitrary colorings. Many new results and proofs have been added, most of which were not known when the first edition was published. Furthermore, the book's tables, exercises, lists of open research problems, and bibliography have all been significantly updated. This innovative book also provides the first cohesive study of Ramsey theory on the integers. It contains perhaps the most substantial account of solved and unsolved problems in this blossoming subject. This breakthrough book will engage students, teachers, and researchers alike.

Essentials of Discrete Mathematics

Author: Richard Gann
Publisher: Jones & Bartlett Publishers
ISBN: 9781284056259
Release Date: 2015-09-01
Genre:

Written for the one-term course, the Third Edition of Essentials of Discrete Mathematics is designed to serve computer science majors as well as students from a wide range of disciplines. The material is organized around five types of thinking: logical, relational, recursive, quantitative, and analytical. This presentation results in a coherent outline that steadily builds upon mathematical sophistication. Graphs are introduced early and referred to throughout the text, providing a richer context for examples and applications. tudents will encounter algorithms near the end of the text, after they have acquired the skills and experience needed to analyze them. The final chapter contains in-depth case studies from a variety of fields, including biology, sociology, linguistics, economics, and music.

Handbook of Mathematical Induction

Author: David S. Gunderson
Publisher: Discrete Mathematics and its Applications
ISBN: 113819901X
Release Date: 2016-10
Genre:

Handbook of Mathematical Induction: Theory and Applications shows how to find and write proofs via mathematical induction. This comprehensive book covers the theory, the structure of the written proof, all standard exercises, and hundreds of application examples from nearly every area of mathematics. In the first part of the book, the author discusses different inductive techniques, including well-ordered sets, basic mathematical induction, strong induction, double induction, infinite descent, downward induction, and several variants. He then introduces ordinals and cardinals, transfinite induction, the axiom of choice, Zorn s lemma, empirical induction, and fallacies and induction. He also explains how to write inductive proofs. The next part contains more than 750 exercises that highlight the levels of difficulty of an inductive proof, the variety of inductive techniques available, and the scope of results provable by mathematical induction. Each self-contained chapter in this section includes the necessary definitions, theory, and notation and covers a range of theorems and problems, from fundamental to very specialized. The final part presents either solutions or hints to the exercises. Slightly longer than what is found in most texts, these solutions provide complete details for every step of the problem-solving process. "

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.

Discrete Mathematics and Its Applications

Author: M. Sethumadhavan
Publisher: Alpha Science Int'l Ltd.
ISBN: 8173197318
Release Date: 2006
Genre: Mathematics

Discrete Mathematics and its Applications provides an in-depth review of recent applications in the area and points to the directions of research. It deals with a wide range of topics like Cryptology Graph Theory Fuzzy Topology Computer Science Mathematical Biology A resource for researchers to keep track of the latest developments in these topics. Of interest to graph theorists, computer scientists, cryptographers, security specialists.

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.

Inevitable Randomness in Discrete Mathematics

Author: J—zsef Beck
Publisher: American Mathematical Soc.
ISBN: 9780821847565
Release Date: 2009-09-01
Genre: Mathematics

Mathematics has been called the science of order. The subject is remarkably good for generalizing specific cases to create abstract theories. However, mathematics has little to say when faced with highly complex systems, where disorder reigns. This disorder can be found in pure mathematical arenas, such as the distribution of primes, the $3n+1$ conjecture, and class field theory. The purpose of this book is to provide examples--and rigorous proofs--of the complexity law: (1) discrete systems are either simple or they exhibit advanced pseudorandomness; (2) a priori probabilities often exist even when there is no intrinsic symmetry. Part of the difficulty in achieving this purpose is in trying to clarify these vague statements. The examples turn out to be fascinating instances of deep or mysterious results in number theory and combinatorics. This book considers randomness and complexity. The traditional approach to complexity--computational complexity theory--is to study very general complexity classes, such as P, NP and PSPACE. What Beck does is very different: he studies interesting concrete systems, which can give new insights into the mystery of complexity. The book is divided into three parts. Part A is mostly an essay on the big picture. Part B is partly new results and partly a survey of real game theory. Part C contains new results about graph games, supporting the main conjecture. To make it accessible to a wide audience, the book is mostly self-contained.