ABSTRACT
Schur’s theorem is one of the earliest results in combinatorial number theory, a mathematical field that is interested in questions that lie between number theory and combinatorics.
This chapter starts with a section about Isaii Schur’s life and work. The following section contains a proof of Schur’s theorem and demonstrates how the theorem, a combinatorial fact, was used to prove seemingly unrelated number theory fact.
The second part of this chapter is devoted to Rado’s theorem, a generalization of Schur’s theorem and another of the milestones in the development of Ramsey theory.
The third section of this chapter provides several biographical facts about Richard Rado and describes some of Rado’s contributions to Ramsey theory. The following section includes a proof of Rado’s theorem.
The end of the chapter contains exercises related to the chapter’s topic.
