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| What This Book is About and How it Came into Being | |
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| Ramsey Theory Before Ramsey, Prehistory and Early History: An Essay in 13 Parts | |
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| Overture | |
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| David Hilbert's 1892 Cube Lemma | |
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| The Issai Schur 1916 Theorem | |
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| The Baudet-Schur-Van der Waerden 1927 Theorem | |
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| The Generalized 1928 Schur Theorem | |
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| The Frank Plumpton Ramsey Principle | |
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| The Paul, Gj�rgy, and Esther Happy End Problem | |
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| Richard Rado's Regularity | |
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| Density and Arithmetic Progressions | |
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| The Tibor Gallai Theorem | |
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| De Bruijn-Erdos's 1951 Compactness Theorem | |
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| Khinchin's Small Book of Big Impact | |
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| Long Live the Young Theory! | |
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| References | |
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| Eighty Years of Ramsey R(3, k) � and Counting! | |
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| Basics | |
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| George, Esther, Paul | |
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| Erdos Magic | |
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| An Erdos Gem | |
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| Upper Bounds | |
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| The Lov�sz Local Lemma | |
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| Random Greedy Triangle-Free | |
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| R(3, k) Resolved! | |
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| Random Greedy Triangle-Free Redux | |
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| Epilogue | |
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| References | |
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| Ramsey Numbers Involving Cycles | |
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| Scope and Notation | |
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| Two-Color Numbers Involving Cycles | |
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| Cycles | |
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| Cycles Versus Complete Graphs | |
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| Cycles Versus Wheels | |
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| Cycles Versus Books | |
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| Cycles Versus Other Graphs | |
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| Multicolor Numbers for Cycles | |
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| Three Colors | |
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| More Colors | |
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| Cycles Versus Other Graphs | |
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| Hypergraph Numbers for Cycles | |
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| References | |
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| On the Function of Erdos and Rogers | |
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| Introduction | |
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| The Most Restrictive Case | |
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| Proof of f<sub>s,s+1</sub>(n)≤0(n<sup>1-1/0(s<sup>4</sup>log s)</sup>)[1] | |
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| Proof of �(n<sup>�</sup>)≤f<sub>s,s+1</sub>(n) for s≥2[2] | |
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| Proof of f<sub>s,s+1</sub>(n) ≤0(n<sup>2/3</sup>) for s≥2 [4] | |
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| General Bounds | |
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| Proof of �(n<sup>a<sub>k</sub>(s)</sup>)≤f<sub>s,s+k</sub>(n) [14, 15] | |
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| Sketch of the Proof of f<sub>s,s+k</sub>(n)≤0(n<sup>((k+1)/(2k+1))+�</sup>) for s≥s<sub>0</sub> = s<sub>0</sub>(�, k) [4] | |
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| Concluding Remarks | |
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| References | |
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| Large Monochromatic Components in Edge Colorings of Graphs: A Survey | |
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| Introduction | |
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| A Remark of Erdos and Rado and Its Extension | |
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| Colorings from Affine Planes | |
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| Extending Colorings by Substitutions | |
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| 2-Colorings | |
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| Type of Spanning Trees, Connectivity, Diameter | |
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| Gallai-Colorings: Substitutions to 2-Colorings | |
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| Multicolorings: Basic Results and Proof Methods | |
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| Complete Bipartite Graphs: Counting Double Stars | |
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| Fractional Transversals: F�redi's Method | |
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| Fine Tuning | |
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| When Both Methods Work: Local Colorings | |
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| Hypergraphs | |
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| Multicolorings: Type of Components | |
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| Components with Large Matching | |
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| Double Stars | |
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| Variations | |
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| Vertex-Coverings by Components | |
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| Coloring by Group Elements | |
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| Coloring Geometric Graphs | |
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| Coloring Noncomplete Graphs | |
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| References | |
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| Szlam's Lemma: Mutant Offspring of a Euclidean Ramsey Problem from 1973, with Numerous Applications | |
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| 1973: A Volcano Erupts | |
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| Some Definitions and More Background | |
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| What Happened to the Rather Red Coloring Problem from 1973? | |
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| Distance Graphs | |
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| Szlam's Lemma, a Connection Between Rather Red Colorings and Chromatic Numbers | |
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| van der Waerden Numbers, Cyclic van der Waerden Numbers, and a Lower Bound on Them Both | |
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| References | |
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| Open Problems in Euclidean Ramsey Theory | |
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| Introduction | |
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| Ramsey Sets | |
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| Unit Distance Graphs | |
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| More General Distance Graphs | |
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| References | |
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| Chromatic Number of the Plane & Its Relatives, History, Problems and Results: An Essay in 11 Parts | |
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| The Problem | |
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| The History | |
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| Polychromatic Number of the Plane & Results Near the Lower Bound | |
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| De Bruijn-Erdos Reduction to Finite Sets and Results Near the Lower Bound | |
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| Polychromatic Number of the Plane & Results Near the Upper Bound | |
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| Continum of 6-Colorings of the Plane | |
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| Chromatic Number of the Plane in Special Circumstances | |
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| Colored Space | |
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| Rational Coloring | |
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| Axioms of Set Theory and the Chromatic Number of Graphs | |
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| Predicting the Future | |
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| References | |
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| Euclidean Distance Graphs on the Rational Points | |
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| Definitions | |
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| The Search for �(R<sup>n</sup>, 1) Leads to the Search for �(Q<sup>n</sup>, 1) | |
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| Distances Other Than 1? | |
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| Problems | |
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| References | |
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| Open Problems Session | |
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| Problems Submitted | |
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| Problems Submitted | |
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| Problems on Topological Stability of Chromatic Numbers Submitted | |
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| Problem on the Gallai-Ramsey Structure, Submitted | |
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| Problems Involving Triangles, Submitted | |
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| Problems on Chromatic Number of the Plane and Its Relatives, Submitted | |
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