Avoidable patterns on two letters

AbstractWe examine avoidable patterns, unavoidable in the sense of Bean, Ehrenfeucht, McNulty (1979). We prove that each pattern on two letters of length at least 13 is avoidable on an alphabet with two letters. The proof is based essentially on two facts: First, each pattern containing an overlapping factor is avoidable by the infinite word of Thue-Morse; secondly, each pattern without overlapping factor is avoidable by the infinite word of Fibonacci. We further discuss the minimal alphabet on which very short patterns on a 2-letter and a 3-letter alphabet are avoidable