Introduction

This chapter is devoted to the study of a special type of unavoidable regularities. We consider a mapping φ:A+ → E from A+ to a set E, and we search in a word w for factors of the type w1w2 … wn with φ(w1) = φ(w2)= … = φ(wn). The mapping is called repetitive when such a factor appears in each sufficiently long word. This is related both to square-free words (Chapter 2), by considering the identity mapping, and to van der Waerden's theorem (Chapter 3), as will be shown later on.

It will first be shown that any mapping from A+ to a finite set is repetitive (Theorem 4.1.1).

After a direct proof of this fact, it will be shown how the result can also be deduced from Ramsey's theorem (which is stated without proof).

Investigated also is the special case where φ is a morphism from A+ to a semigroup S. First it is proved that a morphism to the semigroup of positive integers is repetitive when the alphabet is finite (Theorem 4.2.1). Then it is proved that a morphism to a finite semigroup is uniformly repetitive, in the sense that the words w1, w2,…, wn/i> in the foregoing definition can be chosen of equal length (Theorem 4.2.2). This is, as will be shown, a generalization of van der Waerden's theorem. Finally, the chapter mentions a number of extensions and other results.