Key problems and conjectures have played an important role in promoting the development of Ramsey theory, a field where great progress has been made during the past two decades, with some old problems solved and many new problems proposed. The present book will be helpful to readers who wish to learn about interesting problems in Ramsey theory, to see how they are interconnected, and then to study them in depth. This book is the first problem book of such scope in Ramsey theory. Many unsolved problems, conjectures and related partial results in Ramsey theory are presented, in areas such as extremal graph theory, additive number theory, discrete geometry, functional analysis, algorithm design, and in other areas. Most presented problems are easy to understand, but they may be difficult to solve. They can be appreciated on many levels and by a wide readership, ranging from undergraduate students majoring in mathematics to research mathematicians. This collection is an essential reference for mathematicians working in combinatorics and number theory, as well as for computer scientists studying algorithms.
Contents
Some definitions and notations
Ramsey theory
Bi-color diagonal classical Ramsey numbers
Paley graphs and lower bounds for R(k, k)
Bi-color off-diagonal classical Ramsey numbers
Multicolor classical Ramsey numbers
Generalized Ramsey numbers
Folkman numbers
The Erdős–Hajnal conjecture
Other Ramsey-type problems in graph theory
On van der Waerden numbers and Szemeredi’s theorem
More problems of Ramsey type in additive number theory
Sidon–Ramsey numbers
Games in Ramsey theory
Local Ramsey theory
Set-coloring Ramsey theory
Other problems and conjectures