Ramsey Theory

1. Front Matter

   Pages i-xiv

   PDF

2. Ramsey Theory Before Ramsey, Prehistory and Early History: An Essay in 13 Parts

   • Alexander Soifer

   Pages 1-26

3. Eighty Years of RamseyR(3, k)…and Counting!

   • Joel Spencer

   Pages 27-39

4. Ramsey Numbers Involving Cycles

   • Stanisław P. Radziszowski

   Pages 41-62

5. On the Function of Erdős and Rogers

   • Andrzej Dudek, Vojtěch Rödl

   Pages 63-76

6. Large Monochromatic Components in Edge Colorings of Graphs: A Survey

   • András Gyárfás

   Pages 77-96

7. Szlam’s Lemma: Mutant Offspring of a Euclidean Ramsey Problem from 1973, with Numerous Applications

   • Jeffrey Burkert, Peter Johnson

   Pages 97-113

8. Open Problems in Euclidean Ramsey Theory

   • Ron Graham, Eric Tressler

   Pages 115-120

9. Chromatic Number of the Plane & Its Relatives, History, Problems and Results: An Essay in 11 Parts

   • Alexander Soifer

   Pages 121-161

10. Euclidean Distance Graphs on the Rational Points

    • Peter Johnson Jr.

    Pages 163-176

11. Open Problems Session

    • Alexander Soifer

    Pages 177-189

Ramsey theory is a relatively “new,” approximately 100 year-old direction of fascinating mathematical thought that touches on many classic fields of mathematics such as combinatorics, number theory, geometry, ergodic theory, topology, combinatorial geometry, set theory, and measure theory. Ramsey theory possesses its own unifying ideas, and some of its results are among the most beautiful theorems of mathematics. The underlying theme of Ramsey theory can be formulated as: any finite coloring of a large enough system contains a monochromatic subsystem of higher degree of organization than the system itself, or as T.S. Motzkin famously put it, absolute disorder is impossible. Ramsey Theory: Yesterday, Today, and Tomorrow explores the theory’s history, recent developments, and some promising future directions through invited surveys written by prominent researchers in the field. The first three surveys provide historical background on the subject; the last three address Euclidean Ramsey theory and related coloring problems. In addition, open problems posed throughout the volume and in the concluding open problem chapter will appeal to graduate students and mathematicians alike.

• Extremal graph theory
• Ramsey Number
• combinatorics
• discrete math
• graph theory

“As we learn from the preface, [Ramsey Theory: Yesterday, Today and Tomorrow] grew out of an intentionally non-traditional conference on Ramsey theory. In accordance with that, the book itself is far from being a traditional textbook or reference book on the subject…We learn far more about the history of Ramsey theory than from other sources…the promise of discussing the future is fulfilled by a very extensive list of open problems contributed by numerous participants. Sometimes it is not even clear what the best way of asking a certain question is, and we are shown the raw form of the problem, just as we would if we participated at a conference…the book is undoubtedly a lot of fun. As one expects from a book on Ramsey theory, it is full of problems that are very easy to formulate, but terribly hard to solve…[the book] could well be used for a course using a seminar format, or for self-study by a graduate student familiarizing himself with the subject.”      —MAA Reviews

• at Colorado Springs, College of Letters, Arts, and Sciences, University of Colorado, Colorado Springs, USA

  Alexander Soifer

Alexander Soifer is a Russian born and educated American mathematician, a professor of mathematics at the University of Colorado, an author of some 200 articles on mathematics, history of mathematics, mathematics education, film reviews, etc. He is Senior Vice President of the World Federation of National Mathematics Competitions, which in 2006 awarded him The Paul Erdos Award. 26 years ago Soifer founded and has since chaired the Colorado Mathematical Olympiad, and served on both the USSR and USA Mathematical Olympiads committees. Soifer's Erdos number is 1.

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