Growth problems for avoidable words

Abstract

A word W is said to avoid a word U if no block (subword, factor) of W is the image of U under a homomorphism of free semigroups without unit. The theory of words avoiding xx (square-free words) has been much studied. The word U is said to be avoidable on n letters if there are arbitrarily long words on an n-letter alphabet that avoid U. If U is avoidable on n letters for some n, let μ( U) be the minimum possible such n. We show that μ( U) has a linear bound in terms of the alphabet size of U. We further show that there exists a word that is avoidable on four letters but not on three letters. Moreover, if U is this word, the number of words of length L on a μ( U)-letter alphabet that avoid U has a polynomial bound in terms of L, so that the question of the existence of such an example is resolved in the affirmative. In contrast, for xx the bound is known to be exponential.