VAN DER WAERDEN’S THEOREM AND AVOIDABILITY IN WORDS

摘要

Independently, Pirillo and Varricchio, Halbeisen and Hungerbuhler ¨ and Freedman considered the following problem, open since 1992: Does there exist an infinite word w over a finite subset of Z such that w contains no two consecutive blocks of the same length and sum? We consider some variations on this problem in the light of van der Waerden’s theorem on arithmetic progressions.