This book is dedicated to problems involving colored objects, and especially to results about the existence of certain exciting and unexpected properties that occur regardless of how these objects are colored. In mathematics, these results comprise the area known as Ramsey Theory, which is the mathematical study of combinatorial objects in which a certain degree of order must occur as the scale of the object becomes large. Ramsey Theory thus includes parts of many fields of mathematics, including combinatorics, geometry, and number theory.
This book addresses both famous problems and their history. The main focus is on the story of the discovery of Ramsey Theory. In addition, the author studies the lives of Issai Schur, Pierre Joseph Henry Baudet and B. L. van der Waerden, often including a personal touch. In particular, the book incorporates the author's correspondence with Van der Waerden, Erdös, Baudet, members of the Schur Circle, and others. It also has some unique photographs of the creators of the mathematics presented herein, from Francis Guthrie to Frank Ramsey.
The book succeeds in grabbing the interest of readers. Soifer is particularly interested in attracting young mathematicians to the fascinating world of Ramsey Theory, full of elegant and easily understandable problems for which no particular mathematical knowledge is necessary, but which are very far from being easily solved. It may well succeed! The fact that the book considers the history surrounding the discovery of such problems also makes it of value to historians of science.
Epigraph: To Paint a Bird by Jacques Prévert.- Foreword by Branko Grünbaum.- Foreword by Peter D. Johnson Jr.- Foreword by Cecil Rousseau.- Greetings to the Reader.- Merry-Go-Round.- A Story of Colored Polygons and Arithmetic Progressions.- Colored Plane: Chromatic Number of the Plane.- Chromatic Number of the Plane: The Problem.- Chromatic Number of the Plane: An Historical Essay.- Polychromatic Number of the Plane & Results near the Lower Bound.- De Bruijn-Erdos Reduction to Finite Sets & Results near the Lower Bound.- Polychromatic Number of the Plane & Results near the Upper Bound.- Continuum of 6-Colorings.- Chromatic Number of the Plane in Special Circumstances.- Measurable Chromatic Number of the Plane.- Coloring in Space.- Rational Coloring.- Coloring Graphs.- Chromatic Number of a Graph.- Dimension of a Graph.- Embedding 4-Chromatic Graphs in the Plane.- Embedding World Records.- Edge Chromatic Number of a Graph.- Carsten Thomassen’s 7-Color Theorem.- Coloring Maps.- How The Four Color Conjecture Was Born.- Victorian Comedy of Errors & Colorful Progress.- Kempe-Heawood’s 5-Color Theorem & Tait’s Equivalence.- The 4-Color Theorem.- The Great Debate.- How does one Color Infinite Maps? A Bagatelle.- Chromatic Number of the Plane Meets Map Coloring: Townsend-Woodall’s 5-Color Theorem.- Colored Graphs.- Paul Erdos.- Proof of De Bruijn-Erdos’s Theorem and Its History.- Edge Colored Graphs: Ramsey and Folkman Numbers.- The Ramsey Principle.- From Pigeonhole Principle to Ramsey Principle.- The Happy End Problem.- The Man behind the Theory: Frank Plumpton Ramsey.- Colored Integers: Ramsey Theory before Ramsey & Its AfterMath.- Ramsey Theory before Ramsey: Hilbert’s 1892 Theorem.- Theory before Ramsey: Schur’s Coloring Solution of a Colored Problem & Its Generalizations.- Ramsey Theory before Ramsey: Van der Waerden Tells the Story of Creation.- Whose Conjecture Did Van der Waerden Prove? Two Lives between Two Wars: Issai Schur and Pierre Joseph Henry Baudet.- Monochromatic Arithmetic Progressions: Life after Van der Waerden.- In search of Van der Waerden: The Nazi Leipzig, 1933-1945.- In search of Van der Waerden: The Post War Amsterdam, 1945.- In search of Van der Waerden: The Unsettling Years, 1946-1951.- Colored Polygons: Euclidean Ramsey Theory.- Monochromatic Polygons in a 2-Colored Plane.- 3-Colored Plane, 2-Colored Space and Ramsey Sets.- Gallai’s Theorem.- Colored Integers in Service of Chromtic Number of the Plane: How O’Donnell Unified Ramsey Theory and No One Noticed.- Application of Baudet-Schur-Van der Waerden’s Theorem.- Applications of Bergelson-Leibman’s and Mordell-Faltings’ Theorems.- Solution of an Erdos Problem: O’Donnell’s Theorem.- Predicting the Future.- What if we had no Choice?.- A Glimpse into the Future: Chromatic Number of the Plane, Theorems and Conjectures.- Imagining the Real, Realizing the Imaginary.- Farewell to the Reader.- Two Celebrated Coloring Problems on the Plane.- Bibliography.- Index of Names.- Index of Terms.- Index of Notations.-