ABSTRACT
This chapter discusses several special cases as well as a general case of Ramsey’s theorem, one of the cornerstones of Ramsey theory.
The first section is devoted to the so-called pigeonhole principle for two reasons. Firstly, the pigeonhole principle is a special case of Ramsey’s theorem. Secondly, the pigeonhole principle is one of the fundamental tools that is used throughout this book.
The second section in this chapter describes a search for solutions to three instances of the so-called dinner party problem. The search leads to the establishment of several values of Ramsey numbers. This opens a window through which the reader will be able to reach out to two fascinating and self-standing statements, which are special cases of Ramsey’s theorem, Ramsey’s theorem for graphs and a case of infinite Ramsey’s theorem. Proofs of those two theorems are presented in the next two sections.
The following section includes a proof of Ramsey’s theorem.
The last section contains exercises based on the material presented in this chapter.
