# On long words avoiding Zimin patterns

## TL;DR

This paper investigates the minimal length of words over finite alphabets that necessarily contain Zimin patterns, establishing new exponential bounds and extending understanding of pattern avoidance in combinatorics on words.

## Contribution

It provides the first exponential lower bound for the minimal length of words encountering Zimin patterns over binary alphabets, improving previous doubly-exponential bounds.

## Key findings

- Established a lower bound for f(n,k) as a tower of n-3 exponentials for k=2
- Proved a doubly-exponential upper bound for abelian pattern encounters
- Extended the understanding of unavoidable patterns in words and their minimal lengths

## 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.