Tower-type bounds for unavoidable patterns in words

A word ω is said to contain the pattern P if there is a way to substitute a nonempty word for each letter in P so that the resulting word is a subword of ω. Bean, Ehrenfeucht, and McNulty and, independently, Zimin characterised the patterns P which are unavoidable, in the sense that any sufficiently long word over a fixed alphabet contains P. Zimin’s characterisation says that a pattern is unavoidable if and only if it is contained in a Zimin word, where the Zimin words are defined by Z_1 = x_1 and Z_n = Z_n − 1x_(n)Z_(n) − 1. We study the quantitative aspects of this theorem, obtaining essentially tight tower-type bounds for the function f(n, q), the least integer such that any word of length f(n, q) over an alphabet of size q contains Z_(n). When n = 3, the first nontrivial case, we determine f(n, q) up to a constant factor, showing that f(3, q) = Θ(2_(q)q!)