| What This Book is About and How it Came into Being | p. vii |
| Ramsey Theory Before Ramsey, Prehistory and Early History: An Essay in 13 Parts | p. 1 |
| Overture | p. 1 |
| David Hilbert's 1892 Cube Lemma | p. 2 |
| The Issai Schur 1916 Theorem | p. 3 |
| The Baudet-Schur-Van der Waerden 1927 Theorem | p. 7 |
| The Generalized 1928 Schur Theorem | p. 8 |
| The Frank Plumpton Ramsey Principle | p. 9 |
| The Paul, Gjörgy, and Esther Happy End Problem | p. 11 |
| Richard Rado's Regularity | p. 14 |
| Density and Arithmetic Progressions | p. 17 |
| The Tibor Gallai Theorem | p. 22 |
| De Bruijn-Erdos's 1951 Compactness Theorem | p. 23 |
| Khinchin's Small Book of Big Impact | p. 23 |
| Long Live the Young Theory! | p. 24 |
| References | p. 25 |
| Eighty Years of Ramsey R(3, k) à and Counting! | p. 27 |
| Basics | p. 27 |
| George, Esther, Paul | p. 28 |
| Erdos Magic | p. 29 |
| An Erdos Gem | p. 30 |
| Upper Bounds | p. 31 |
| The Lovász Local Lemma | p. 33 |
| Random Greedy Triangle-Free | p. 34 |
| R(3, k) Resolved! | p. 35 |
| Random Greedy Triangle-Free Redux | p. 36 |
| Epilogue | p. 38 |
| References | p. 38 |
| Ramsey Numbers Involving Cycles | p. 41 |
| Scope and Notation | p. 41 |
| Two-Color Numbers Involving Cycles | p. 42 |
| Cycles | p. 43 |
| Cycles Versus Complete Graphs | p. 44 |
| Cycles Versus Wheels | p. 46 |
| Cycles Versus Books | p. 47 |
| Cycles Versus Other Graphs | p. 49 |
| Multicolor Numbers for Cycles | p. 50 |
| Three Colors | p. 50 |
| More Colors | p. 53 |
| Cycles Versus Other Graphs | p. 54 |
| Hypergraph Numbers for Cycles | p. 55 |
| References | p. 56 |
| On the Function of Erdos and Rogers | p. 63 |
| Introduction | p. 63 |
| The Most Restrictive Case | p. 64 |
| Proof of fs,s+1(n)≤0(n1-1/0(s4log s))[1] | p. 65 |
| Proof of ¿(n½)≤fs,s+1(n) for s≥2[2] | p. 67 |
| Proof of fs,s+1(n) ≤0(n2/3) for s≥2 [4] | p. 67 |
| General Bounds | p. 69 |
| Proof of ¿(nak(s))≤fs,s+k(n) [14, 15] | p. 71 |
| Sketch of the Proof of fs,s+k(n)≤0(n((k+1)/(2k+1))+¿) for s≥s0 = s0(¿, k) [4] | p. 73 |
| Concluding Remarks | p. 75 |
| References | p. 75 |
| Large Monochromatic Components in Edge Colorings of Graphs: A Survey | p. 77 |
| Introduction | p. 77 |
| A Remark of Erdos and Rado and Its Extension | p. 77 |
| Colorings from Affine Planes | p. 78 |
| Extending Colorings by Substitutions | p. 79 |
| 2-Colorings | p. 79 |
| Type of Spanning Trees, Connectivity, Diameter | p. 79 |
| Gallai-Colorings: Substitutions to 2-Colorings | p. 82 |
| Multicolorings: Basic Results and Proof Methods | p. 83 |
| Complete Bipartite Graphs: Counting Double Stars | p. 83 |
| Fractional Transversals: Füredi's Method | p. 85 |
| Fine Tuning | p. 85 |
| When Both Methods Work: Local Colorings | p. 86 |
| Hypergraphs | p. 88 |
| Multicolorings: Type of Components | p. 89 |
| Components with Large Matching | p. 89 |
| Double Stars | p. 90 |
| Variations | p. 90 |
| Vertex-Coverings by Components | p. 91 |
| Coloring by Group Elements | p. 91 |
| Coloring Geometric Graphs | p. 91 |
| Coloring Noncomplete Graphs | p. 92 |
| References | p. 94 |
| Szlam's Lemma: Mutant Offspring of a Euclidean Ramsey Problem from 1973, with Numerous Applications | p. 97 |
| 1973: A Volcano Erupts | p. 97 |
| Some Definitions and More Background | p. 98 |
| What Happened to the Rather Red Coloring Problem from 1973? | p. 101 |
| Distance Graphs | p. 102 |
| Szlam's Lemma, a Connection Between Rather Red Colorings and Chromatic Numbers | p. 104 |
| van der Waerden Numbers, Cyclic van der Waerden Numbers, and a Lower Bound on Them Both | p. 107 |
| References | p. 112 |
| Open Problems in Euclidean Ramsey Theory | p. 115 |
| Introduction | p. 115 |
| Ramsey Sets | p. 117 |
| Unit Distance Graphs | p. 117 |
| More General Distance Graphs | p. 119 |
| References | p. 119 |
| Chromatic Number of the Plane & Its Relatives, History, Problems and Results: An Essay in 11 Parts | p. 121 |
| The Problem | p. 121 |
| The History | p. 124 |
| Polychromatic Number of the Plane & Results Near the Lower Bound | p. 135 |
| De Bruijn-Erdos Reduction to Finite Sets and Results Near the Lower Bound | p. 138 |
| Polychromatic Number of the Plane & Results Near the Upper Bound | p. 141 |
| Continum of 6-Colorings of the Plane | p. 144 |
| Chromatic Number of the Plane in Special Circumstances | p. 148 |
| Colored Space | p. 149 |
| Rational Coloring | p. 151 |
| Axioms of Set Theory and the Chromatic Number of Graphs | p. 153 |
| Predicting the Future | p. 156 |
| References | p. 157 |
| Euclidean Distance Graphs on the Rational Points | p. 163 |
| Definitions | p. 163 |
| The Search for ¿(Rn, 1) Leads to the Search for ¿(Qn, 1) | p. 164 |
| Distances Other Than 1? | p. 167 |
| Problems | p. 172 |
| References | p. 174 |
| Open Problems Session | p. 177 |
| Problems Submitted | p. 179 |
| Problems Submitted | p. 180 |
| Problems on Topological Stability of Chromatic Numbers Submitted | p. 182 |
| Problem on the Gallai-Ramsey Structure, Submitted | p. 183 |
| Problems Involving Triangles, Submitted | p. 185 |
| Problems on Chromatic Number of the Plane and Its Relatives, Submitted | p. 189 |
| Table of Contents provided by Ingram. All Rights Reserved. |
