The Mathematical Coloring Book

This is a unique type of book; at least, I have never encountered a book of this kind. The best description of it I can give is that it is a mystery novel, developing on three levels, and imbued with both educational and philosophical/moral issues. If this summary description does not help understanding the particular character and allure of the book, possibly a more detailed explanation will be found useful. One of the primary goals of the author is to interest readers—in particular, young mathematiciansorpossiblypre-mathematicians—inthefascinatingworldofelegant and easily understandable problems, for which no particular mathematical kno- edge is necessary, but which are very far from being easily solved. In fact, the prototype of such problems is the following: If each point of the plane is to be given a color, how many colors do we need if every two points at unit distance are to receive distinct colors? More than half a century ago it was established that the least number of colors needed for such a coloring is either 4, or 5, or 6 or 7. Well, which is it? Despite efforts by a legion of very bright people—many of whom developed whole branches of mathematics and solved problems that seemed much harder—not a single advance towards the answer has been made. This mystery, and scores of other similarly simple questions, form one level of mysteries explored. In doing this, the author presents a whole lot of attractive results in an engaging way, and with increasing level of depth.

This is a unique type of book; at least, I have never encountered a book of this kind. The best description of it I can give is that it is a mystery novel, developing on three levels, and imbued with both educational and philosophical/moral issues. If this summary description does not help understanding the particular character and allure of the book, possibly a more detailed explanation will be found useful. One of the primary goals of the author is to interest readers—in particular, young mathematiciansorpossiblypre-mathematicians—inthefascinatingworldofelegant and easily understandable problems, for which no particular mathematical kno- edge is necessary, but which are very far from being easily solved. In fact, the prototype of such problems is the following: If each point of the plane is to be given a color, how many colors do we need if every two points at unit distance are to receive distinct colors? More than half a century ago it was established that the least number of colorsneeded for such a coloring is either 4, or 5, or 6 or 7. Well, which is it? Despite efforts by a legion of very bright people—many of whom developed whole branches of mathematics and solved problems that seemed much harder—not a single advance towards the answer has been made. This mystery, and scores of other similarly simple questions, form one level of mysteries explored. In doing this, the author presents a whole lot of attractive results in an engaging way, and with increasing level of depth.

Merry-Go-Round.- A Story of Colored Polygons and Arithmetic Progressions.- Colored Plane.- Chromatic Number of the Plane: The Problem.- Chromatic Number of the Plane: An Historical Essay.- Polychromatic Number of the Plane and Results Near the Lower Bound.- De Bruijn–Erd?s Reduction to Finite Sets and Results Near the Lower Bound.- Polychromatic Number of the Plane and Results Near the Upper Bound.- Continuum of 6-Colorings of the Plane.- 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 and Colorful Progress.- Kempe–Heawood’s Five-Color Theorem and Tait’s Equivalence.- The Four-Color Theorem.- The GreatDebate.- 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 Erd?s.- De Bruijn–Erd?s’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 and Its AfterMath.- Ramsey Theory Before Ramsey: Hilbert’s Theorem.- Ramsey Theory Before Ramsey: Schur’s Coloring Solution of a Colored Problem and 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 Early Years.- In Search of Van der Waerden: The Nazi Leipzig, 1933–1945.- In Search of Van der Waerden: The Postwar Amsterdam, 1945166.- 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 Chromatic Number of the Plane: How O’Donnell Unified Ramsey Theory and No One Noticed.- Application of Baudet–Schur–Van der Waerden.- Application of Bergelson–Leibman’s and Mordell–Faltings’ Theorems.- Solution of an Erd?s 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 Problems.