  A word $w$ 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 $w$. 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 $Z_n=Z_{n-1} x_n Z_{n-1}$. 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 non-trivial case, we determine $f(n,q)$ up to a constant factor,
showing that $f(3,q) = \Theta(2^q q!)$.
