Download or read book Adventures in Graph Ramsey Theory written by Andrew T. Parrish and published by . This book was released on 2013 with total page 68 pages. Available in PDF, EPUB and Kindle. Book excerpt: We define what it means for an equation to be graph-regular, extending the idea of partition-regular equations to a graph setting. An equation is graph-regular if it always has monochromatic solutions under edge-colorings of K/N. We find an infinite family of graph-regular equations, and present two Rado-like conditions which are respectively necessary and sufficient for an equation to be graph-regular. In the process, we prove a Ramsey-like theorem for binary and k-ary trees which may be of independent interest. We also look at a stronger version of Ramsey's theorem from Paris and Harrington, and show a counterexample to the analogous version of van der Waerden's theorem.
Download or read book Elementary Methods of Graph Ramsey Theory written by Yusheng Li and published by Springer Nature. This book was released on 2022-09-16 with total page 349 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended to provide graduate students and researchers in graph theory with an overview of the elementary methods of graph Ramsey theory. It is especially targeted towards graduate students in extremal graph theory, graph Ramsey theory, and related fields, as the included contents allow the text to be used in seminars. It is structured in thirteen chapters which are application-focused and largely independent, enabling readers to target specific topics and information to focus their study. The first chapter includes a true beginner’s overview of elementary examples in graph Ramsey theory mainly using combinatorial methods. The following chapters progress through topics including the probabilistic methods, algebraic construction, regularity method, but that's not all. Many related interesting topics are also included in this book, such as the disproof for a conjecture of Borsuk on geometry, intersecting hypergraphs, Turán numbers and communication channels, etc.
Download or read book Erd s on Graphs written by Fan Chung and published by CRC Press. This book was released on 2020-08-26 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a tribute to Paul Erdos, the wandering mathematician once described as the "prince of problem solvers and the absolute monarch of problem posers." It examines the legacy of open problems he left to the world after his death in 1996.
Download or read book The Triangle Free Process and the Ramsey Number R 3 k written by Gonzalo Fiz Pontiveros and published by American Mathematical Soc.. This book was released on 2020-04-03 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt: The areas of Ramsey theory and random graphs have been closely linked ever since Erdős's famous proof in 1947 that the “diagonal” Ramsey numbers R(k) grow exponentially in k. In the early 1990s, the triangle-free process was introduced as a model which might potentially provide good lower bounds for the “off-diagonal” Ramsey numbers R(3,k). In this model, edges of Kn are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted Gn,△. In 2009, Bohman succeeded in following this process for a positive fraction of its duration, and thus obtained a second proof of Kim's celebrated result that R(3,k)=Θ(k2/logk). In this paper the authors improve the results of both Bohman and Kim and follow the triangle-free process all the way to its asymptotic end.
Download or read book Ramsey Theory written by Xiaodong Xu and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-08-06 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: Key problems and conjectures have played an important role in promoting the development of Ramsey theory, a field where great progress has been made during the past two decades, with some old problems solved and many new problems proposed. The present book will be helpful to readers who wish to learn about interesting problems in Ramsey theory, to see how they are interconnected, and then to study them in depth. This book is the first problem book of such scope in Ramsey theory. Many unsolved problems, conjectures and related partial results in Ramsey theory are presented, in areas such as extremal graph theory, additive number theory, discrete geometry, functional analysis, algorithm design, and in other areas. Most presented problems are easy to understand, but they may be difficult to solve. They can be appreciated on many levels and by a wide readership, ranging from undergraduate students majoring in mathematics to research mathematicians. This collection is an essential reference for mathematicians working in combinatorics and number theory, as well as for computer scientists studying algorithms. Contents Some definitions and notations Ramsey theory Bi-color diagonal classical Ramsey numbers Paley graphs and lower bounds for R(k, k) Bi-color off-diagonal classical Ramsey numbers Multicolor classical Ramsey numbers Generalized Ramsey numbers Folkman numbers The Erdős–Hajnal conjecture Other Ramsey-type problems in graph theory On van der Waerden numbers and Szemeredi’s theorem More problems of Ramsey type in additive number theory Sidon–Ramsey numbers Games in Ramsey theory Local Ramsey theory Set-coloring Ramsey theory Other problems and conjectures
Download or read book Topics in Gallai Ramsey Theory written by Colton Magnant and published by Springer Nature. This book was released on 2020-07-04 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores topics in Gallai-Ramsey theory, which looks into whether rainbow colored subgraphs or monochromatic subgraphs exist in a sufficiently large edge-colored complete graphs. A comprehensive survey of all known results with complete references is provided for common proof methods. Fundamental definitions and preliminary results with illustrations guide readers to comprehend recent innovations. Complete proofs and influential results are discussed with numerous open problems and conjectures. Researchers and students with an interest in edge-coloring, Ramsey Theory, and colored subgraphs will find this book a valuable guide for entering Gallai-Ramsey Theory.
Download or read book A Generalization of Ramsey Theory for Graphs written by Chung Laung Liu and published by . This book was released on 1977 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Ramsey Theory written by Henry Glynn Carnick and published by . This book was released on 2018 with total page 86 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Topics in finite graph Ramsey theory written by and published by . This book was released on 2001 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: February 2008.
Download or read book Some results in graph Ramsey theory and graph representations written by Nancy Eaton and published by . This book was released on 1992 with total page 91 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book On Ramsey Numbers written by Kashif Ali and published by . This book was released on 2009-12 with total page 72 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Ramsey Theory for Graphs written by Sharon Rogolsky and published by . This book was released on 1976 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Some Problems in Graph Ramsey Theory written by Andrey Vadim Grinshpun and published by . This book was released on 2015 with total page 156 pages. Available in PDF, EPUB and Kindle. Book excerpt: A graph G is r-Ramsey minimal with respect to a graph H if every r-coloring of the edges of G yields a monochromatic copy of H, but the same is not true for any proper subgraph of G. The study of the properties of graphs that are Ramsey minimal with respect to some H and similar problems is known as graph Ramsey theory; we study several problems in this area. Burr, Erdös, and Lovász introduced s(H), the minimum over all G that are 2- Ramsey minimal for H of [delta](G), the minimum degree of G. We find the values of s(H) for several classes of graphs H, most notably for all 3-connected bipartite graphs which proves many cases of a conjecture due to Szabó, Zumstein, and Zürcher. One natural question when studying graph Ramsey theory is what happens when, rather than considering all 2-colorings of a graph G, we restrict to a subset of the possible 2-colorings. Erdös and Hajnal conjectured that, for any fixed color pattern C, there is some [epsilon] > 0 so that every 2-coloring of the edges of a Kn, the complete graph on n vertices, which doesn't contain a copy of C contains a monochromatic clique on n[epsilon] vertices. Hajnal generalized this conjecture to more than 2 colors and asked in particular about the case when the number of colors is 3 and C is a rainbow triangle (a K3 where each edge is a different color); we prove Hajnal's conjecture for rainbow triangles. One may also wonder what would happen if we wish to cover all of the vertices with monochromatic copies of graphs. Let F = {F1, F2, . . .} be a sequence of graphs such that Fn is a graph on n vertices with maximum degree at most [delta]. If each Fn is bipartite, then the vertices of any 2-edge-colored complete graph can be partitioned into at most 2C[delta] vertex disjoint monochromatic copies of graphs from F, where C is an absolute constant. This result is best possible, up to the constant C.
Download or read book An Introduction to Ramsey Theory Fast Functions Infinity and Metamathematics written by Matthew Katz and published by American Mathematical Soc.. This book was released on 2018-10-03 with total page 207 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book takes the reader on a journey through Ramsey theory, from graph theory and combinatorics to set theory to logic and metamathematics. Written in an informal style with few requisites, it develops two basic principles of Ramsey theory: many combinatorial properties persist under partitions, but to witness this persistence, one has to start with very large objects. The interplay between those two principles not only produces beautiful theorems but also touches the very foundations of mathematics. In the course of this book, the reader will learn about both aspects. Among the topics explored are Ramsey's theorem for graphs and hypergraphs, van der Waerden's theorem on arithmetic progressions, infinite ordinals and cardinals, fast growing functions, logic and provability, Gödel incompleteness, and the Paris-Harrington theorem. Quoting from the book, “There seems to be a murky abyss lurking at the bottom of mathematics. While in many ways we cannot hope to reach solid ground, mathematicians have built impressive ladders that let us explore the depths of this abyss and marvel at the limits and at the power of mathematical reasoning at the same time. Ramsey theory is one of those ladders.”
Download or read book The Mathematical Coloring Book written by Alexander Soifer and published by Springer Science & Business Media. This book was released on 2008-10-13 with total page 619 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an exciting history of the discovery of Ramsey Theory, and contains new research along with rare photographs of the mathematicians who developed this theory, including Paul Erdös, B.L. van der Waerden, and Henry Baudet.
Download or read book Results in Extremal Graph Theory Ramsey Theory and Additive Combinatorics written by Oliver Janzer and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Combinatorics and Graph Theory written by John Harris and published by Springer Science & Business Media. This book was released on 2009-04-03 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes were first used in an introductory course team taught by the authors at Appalachian State University to advanced undergraduates and beginning graduates. The text was written with four pedagogical goals in mind: offer a variety of topics in one course, get to the main themes and tools as efficiently as possible, show the relationships between the different topics, and include recent results to convince students that mathematics is a living discipline.
