Exponential lower bounds for the number of words of uniform length avoiding a pattern

We study words on a finite alphabet avoiding a finite collection of patterns. Given a pattern p in which every letter that occurs in p occurs at least twice, we show that the number of words of length n on a finite alphabet that avoid p grows exponentially with n as long as the alphabet has at least four letters. Moreover, we give lower bounds describing this exponential growth in terms of the size of the alphabet and the number of letters occurring in p. We also obtain analogous results for the number of words avoiding a finite collection of patterns. We conclude by giving some questions.