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.

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http://dx.doi.org/10.1090/stml/073/01