On long words avoiding Zimin patterns

Abstract: A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern $p$ is unavoidable if, over every finite alphabet, every sufficiently long word encounters $p$. A theorem by Zimin and independently by Bean, Ehrenfeucht and McNulty states that a pattern over $n$ distinct variables is unavoidable if, and only if, $p$ itself is encountered in the $n$-th Zimin pattern. Given an alphabet size $k$, we study the minimal length $f(n,k)$ such that every word of length $f(n,k)$ encounters the $n$-th Zimin pattern. It is known that $f$ is upper-bounded by a tower of exponentials. Our main result states that $f(n,k)$ is lower-bounded by a tower of $n-3$ exponentials, even for $k=2$. To the best of our knowledge, this improves upon a previously best-known doubly-exponential lower bound. As a further result, we prove a doubly-exponential upper bound for encountering Zimin patterns in the abelian sense.