''The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators'' is a book on

graph coloring In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a Graph (discrete mathematics), graph. The assignment is subject to certain constraints, such as that no two adjacent elements have th ...

Ramsey theory Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in R ...

, and the history of development of these areas, concentrating in particular on the

Hadwiger–Nelson problem In geometric graph theory, the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson, asks for the minimum number of colors required to color the plane such that no two points at distance 1 from each other have the same color. ...

and on the biography of

Bartel Leendert van der Waerden Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics. Biography Education and early career Van der Waerden learned advanced mathematics at the University of Amste ...

. It was written by

Alexander Soifer Alexander Soifer is a Russian-born American mathematician and mathematics author. Soifer obtained his Ph.D. in 1973 and has been a professor of mathematics at the University of Colorado since 1979. He was visiting fellow at Princeton University ...

and published by

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

in 2009 (). The book has since been updated:
The New Mathematical Coloring Book
Mathematics of Coloring and the Colorful Life of Its Creators'' was published in 2024.

Topics

The book "presents mathematics as a human endeavor" and "explores the birth of ideas and moral dilemmas of the times between and during the two World Wars". As such, as well as covering the mathematics of its topics, it includes biographical material and correspondence with many of the people involved in creating it, including in-depth coverage of

Issai Schur Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the Humboldt University of Berlin, University of Berlin. He obtained his doctorate in 1901, became lecturer i ...

, , and

, in particular studying the question of van der Warden's complicity with the Nazis in his war-time service as a professor in Nazi Germany. It also includes biographical material on

Paul Erdős Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, g ...

Frank P. Ramsey Frank Plumpton Ramsey (; 22 February 1903 – 19 January 1930) was a British people, British philosopher, mathematician, and economist who made major contributions to all three fields before his death at the age of 26. He was a close friend of ...

Emmy Noether Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...

Alfred Brauer Alfred Theodor Brauer (April 9, 1894 – December 23, 1985) was a German-American mathematician who did work in number theory. He was born in Charlottenburg, and studied at the Humboldt University of Berlin, University of Berlin. As he served ...

Richard Courant Richard Courant (January 8, 1888 – January 27, 1972) was a German-American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real ...

, Kenneth Falconer, Nicolas de Bruijn,

Hillel Furstenberg Hillel "Harry" Furstenberg (; born September 29, 1935) is a German-born American-Israeli mathematician and professor emeritus at the Hebrew University of Jerusalem. He is a member of the Israel Academy of Sciences and Humanities and U.S. Natio ...

, and

Tibor Gallai Tibor Gallai (born Tibor Grünwald, 15 July 1912 – 2 January 1992) was a Hungarian mathematician. He worked in combinatorics, especially in graph theory, and was a lifelong friend and collaborator of Paul Erdős. He was a student of Dénes K� ...

, among others, as well as many historical photos of these subjects. Mathematically, the book considers problems "on the boundary of geometry, combinatorics, and number theory", involving

problems such as the

four color theorem In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions shar ...

, and generalizations of coloring in

where the use of a too-small number of colors leads to monochromatic structures larger than a single graph edge. Central to the book is the

, the problem of coloring the points of the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

in such a way that no two points of the same color are a unit distance apart. Other topics covered by the book include

Van der Waerden's theorem Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory. Van der Waerden's theorem states that for any given positive integers ''r'' and ''k'', there is some number ''N'' such that if the integers are colored, eac ...

on monochromatic

arithmetic progression An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...

s in colorings of the integers and its generalization to

Szemerédi's theorem In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers ''A'' with positive natural density contains a ''k''- ...

, the

Happy ending problem In mathematics, the "happy ending problem" (so named by Paul Erdős because it led to the marriage of George Szekeres and Esther Szekeres, Esther Klein) is the following statement: This was one of the original results that led to the develop ...

, Rado's theorem, and questions in the

foundations of mathematics Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theo ...

involving the possibility that different choices of foundational axioms will lead to different answers to some of the coloring questions considered here.

Reception and audience

As a work in

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, reviewer Joseph Malkevitch suggests caution over the book's intuitive treatment of graphs that may in many cases be infinite, in comparison with much other work in this area that makes an implicit assumption that every graph is finite.

William Gasarch William Ian Gasarch ( ; born 1959) is an American computer scientist known for his work in computational complexity theory, computability theory, computational learning theory, and Ramsey theory. He is currently a professor at the University of M ...

is surprised by the book's omission of some closely related topics, including the proof of the

Heawood conjecture In graph theory, the Heawood conjecture or Ringel–Youngs theorem gives a lower bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element o ...

on coloring graphs on surfaces by

Gerhard Ringel Gerhard Ringel (October 28, 1919 in Kollnbrunn, Austria – June 24, 2008 in Santa Cruz, California) was a German mathematician. He was one of the pioneers in graph theory and contributed significantly to the proof of the Heawood conjecture (now ...

and Ted Youngs. And Günter M. Ziegler complains that many claims are presented without proof. Although Soifer has called the Hadwiger–Nelson problem "the most important problem in all of mathematics", Ziegler disagrees, and suggests that it and the four color theorem are too isolated to be fruitful topics of study. As a work in the

history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples ...

, Malkevitch finds the book too credulous of first-person recollections of troubled political times (the lead-up to

World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...

) and of priority in mathematical discoveries. Ziegler points to several errors of fact in the book's history, takes issue with its insistence that each contribution should be attributed to only one researcher, and doubts Soifer's objectivity with respect to van der Waerden. And reviewer John J. Watkins writes that "Soifer’s book is indeed a treasure trove filled with valuable historical and mathematical information, but a serious reader must also be prepared to sift through a considerable amount of dross" to reach the treasure. And although Watkins is convinced by Soifer's argument that the first conjectural versions of van der Waerden's theorem were due to Schur and Baudet, he finds idiosyncratic Soifer's insistence that this updated credit necessitates a change in the name of the theorem, concluding that "This is a book that needed far better editing." Ziegler agrees, writing "Someone should have also forced him to cut the manuscript, at the long parts and chapters where the investigations into the colorful lives of the creators get out of hand." According to Malkevitch, the book is written for a broad audience, and does not require a graduate-level background in its material, but nevertheless contains much that is of interest to experts as well as beginners. And despite his negative review, Ziegler concurs, writing that it "has interesting parts and a lot of valuable material". Gasarch is much more enthusiastic, writing "This is a Fantastic Book! Go buy it Now!".

References

{{DEFAULTSORT:Mathematical Coloring Book, The Graph coloring Ramsey theory Books about the history of mathematics 2009 non-fiction books