It is no exaggeration to say that over the past several
decades there has been a veritable explosion of activity in the
general field of combinatorics. Ramsey theory, in particular, has
shown remarkable growth. This book gives a picture of the state of
the art of Ramsey theory at the time of Graham's CBMS lectures. In
keeping with the style of the lectures, the exposition is
informal. However, complete proofs are given for most of the basic
results presented. In addition, many useful results may be found in
the exercises and problems.
Loosely speaking, Ramsey theory is the branch of combinatorics that
deals with structures that are preserved under partitions. Typically,
one looks at the following kind of question: If a particular structure
(e.g., algebraic, combinatorial or geometric) is arbitrarily
partitioned into finitely many classes, what kinds of substructures
must always remain intact in at least one of the classes?
At the time of these lectures, a number of spectacular advances had
been made in the field of Ramsey theory. These include: the work of
Szemerédi and Furstenberg settling the venerable conjecture of
Erdős and Turán, the Nešetril-Rödl theorems on
induced Ramsey properties, the results of Paris and Harrington on
“large” Ramsey numbers and undecidability in first-order
Peano arithmetic, Deuber's solution to the old partition regularity
conjecture of Rado, Hindman's surprising generalization of Schur's
theorem, and the resolution of Rota's conjecture on Ramsey's theorem
for vector spaces by Graham, Leeb and Rothschild. It has also become
apparent that the ideas and techniques of Ramsey theory span a rather
broad range of mathematical areas, interacting in essential ways with
parts of set theory, graph theory, combinatorial number theory,
probability theory, analysis and even theoretical computer
science. These lecture notes lay out the foundation on which much of
this work is based.
Relatively little specialized mathematical background is required
for this book. It should be accessible to upper division
students.
Readership
Advanced undergraduates, graduate students, and
post-Ph.D. mathematicians interested in combinatorics.