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li y elementary methods graph ramsey theory 2022

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Textbook in PDF format 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 Preface Existence Terminology General Upper Bounds Upper Bounds for rk(3) Some Early Ramsey Numbers Hypergraph Ramsey Number Exercises Small Ramsey Numbers Ramsey Folklore Finite Field and r3(3) Schur Numbers Paley Graphs Combination of Paley Graphs Spectrum and Independence Number Exercises Basic Probabilistic Method Some Basic Inequalities A Lower Bound of r(n,n) Pick Vertices Semi-Randomly Independence Number of Sparse Graphs Upper Bounds for r(m,n) Odd Cycle versus Large Kn The First Two Moments Chernoff Bounds Exercises Random Graph Preliminary Lower Bounds for r(m,n) More Applications of Chernoff Bounds Properties of Random Graphs Some Behaviors of Almost All Graphs Parameters of Random Graphs Threshold Functions Poisson Limit Exercises Lovász Local Lemma Lovász Local Lemma Improved Lower Bounds for r(m,n) Martingales and Triangle-Free Process Exercises Constructive Lower Bounds Constructive Lower Bounds for r(s,t) Constructive Lower Bounds for r(t) A Conjecture of Borsuk Intersecting Hypergraphs Lower Bounds of rk(t) for k3 Exercises Turán Number and Related Ramsey Number Turán Numbers for Non-Bipartite Graphs Turán Numbers for Kt,s Erdős-Rényi Graph Exact Values of ex(n,C4) and z(n;2) Constructions with Forbidden K2,s Constructions with Forbidden Kt,s Turán Numbers for Even Cycles Exercises Communication Channels Introduction Shannon Capacities of Cycles Connection with Ramsey Numbers Exercises Dependent Random Choice The Basic Lemma Applications Exercises Quasi-Random Graphs Properties of Dense Graphs Graphs with Small Second Eigenvalues Some Multicolor Ramsey Numbers A Related Lower Bound of r(s,t) A Lower Bound for Book Graph Exercises Regularity Lemma and van der Waerden Number van der Waerden Number Recursive Bounds for wk(t) Szemerédi's Regularity Lemma Two Applications Extensions on the Regularity Lemma Exercises More Ramsey Linear Functions Subdivided Graphs Ramsey Goodness Large Books Are p-Good Exercises Various Ramsey Problems Size Ramsey Numbers Induced Ramsey Numbers Bipartite Ramsey Numbers Folkman Numbers For Parameters and Coloring Types Exercises References Glossary Index

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li y elementary methods graph ramsey theory 2022

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