Chapter 1
Preliminaries
Unsolved problems abound, and additional inter-
esting open questions arise faster than solutions to
the existing problems. F. Harary
The above quote, which appeared in the 1983 article “A Trib-
ute to F. P. Ramsey,” is at least as apropos today as it was then.
In this book alone, which covers only a modest portion of Ramsey
theory, you will find a great number of open research problems. The
beauty of Ramsey theory, especially Ramsey theory dealing with the
set of integers, is that, unlike many other mathematical fields, very
little background is needed to understand the problems. In fact, with
just a basic understanding of some of the topics in this text, and a
desire to discover new results, the undergraduate mathematics stu-
dent will be able to experience the excitement and challenge of doing
mathematical research.
Ramsey theory is named after Frank Plumpton Ramsey and his
eponymous theorem, which he proved in 1928 (it was published post-
humously in 1930). So, what is Ramsey theory? Although there
is no universally accepted definition of Ramsey theory, we offer the
following informal description:
Ramsey theory is the study of the preservation of
properties under set partitions.
1
http://dx.doi.org/10.1090/stml/073/01
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