Ramsey theory yesterday, today, and tomorrow ; [the workshop took place on May 27-29, 2009 at the Busch Campus of Rutgers University in Piscataway, New Jersey

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. Contributors: J. Burkert, A. Dudek, R.L. Graham, A. Gyárfás, P.D. Johnson, Jr., S.P. Radziszowski, V. Rödl, J.H. Spencer, A. Soifer, E. Tressler. 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. Contributors: J. Burkert, A. Dudek, R.L. Graham, A. Gyárfás, P.D. Johnson, Jr., S.P. Radziszowski, V. Rödl, J.H. Spencer, A. Soifer, E. Tressler Ramsey Theory Before Ramsey, Prehistory And Early History: An Essay In 13 Parts / Alexander Soifer -- Eighty Years Of Ramsey R (3,k)...and Counting! / Joel Spencer -- Ramsey Numbers Involving Cycles / Stanislaw P. Radziszowski -- On The Function Of Erdős And Rogers / Andrzej Dudek And Vojtĕch Rödl -- Large Monochromatic Components In Edge Colorings Of Graphs: A Survey / András Gyárfás -- Szlam's Lemma: Mutant Offspring Of A Euclidean Ramsey Problem From 1973, With Numerous Applications / Jeffrey Burkert And Peter Johnson, Jr. -- Open Problems In Euclidean Ramsey Theory / Ron Graham And Eric Tressler -- Chromatic Number Of The Plane & Its Relatives, History, Problems And Results: An Essay In 11 Parts / Alexander Soifer -- Euclidean Distance Graphs On The Rational Points / Peter Johnson, Jr. -- Open Problems Session. Alexander Soifer, Editor. Includes Bibliographical References. Front Matter....Pages i-xiv Ramsey Theory Before Ramsey, Prehistory and Early History: An Essay in 13 Parts....Pages 1-26 Eighty Years of Ramsey R (3, k )...and Counting!....Pages 27-39 Ramsey Numbers Involving Cycles....Pages 41-62 On the Function of Erdős and Rogers....Pages 63-76 Large Monochromatic Components in Edge Colorings of Graphs: A Survey....Pages 77-96 Szlam’s Lemma: Mutant Offspring of a Euclidean Ramsey Problem from 1973, with Numerous Applications....Pages 97-113 Open Problems in Euclidean Ramsey Theory....Pages 115-120 Chromatic Number of the Plane & Its Relatives, History, Problems and Results: An Essay in 11 Parts....Pages 121-161 Euclidean Distance Graphs on the Rational Points....Pages 163-176 Open Problems Session....Pages 177-189