Long unavoidable patterns

Summary We examine long unavoidable patterns, unavoidable in the sense of Bean, Ehrenfeucht, McNulty. Zimin and independently Schmidt have shown that there is only one unavoidable pattern of length 2 ⁿ -1 on an alphabet with n letters; this pattern is a quasi-power in the sense of Schützenberger. We characterize the unavoidable words of length 2 ⁿ -2 and 2 ⁿ -3. Finally we show that every sufficiently long unavoidable word has a certain quasi-power as a subword.This work was done while the author stayed at LITP, Université Paris 6, France