An Application of van der Waerden's Theorem in Additive Number Theory

A sequence on a finite set of symbols is called strongly non-repetitive if no two adjacent (finite) segments are permutations of each other. Replacing the finite set of symbols of a strongly non-repetitive sequence by different prime numbers, one gets an infinite sequence on a finite set of integers such that no two adjacent segments have the same product. It is known that there are infinite strongly non-repetitive sequences on just four symbols. The aim of this paper is to show that there is no infinite sequence on a finite set of integers such that no two adjacent segments have the same sum. Thus, in the statement above, one cannot replace “product” by “sum”. Further we suggest some strengthened versions of the notion of strongly non-repetitive