# Tower-type bounds for unavoidable patterns in words

## TL;DR

This paper investigates the bounds on the length of words over finite alphabets that guarantee the presence of unavoidable patterns, specifically Zimin patterns, establishing tight tower-type bounds for these functions.

## Contribution

It provides essentially tight tower-type bounds for the function determining the minimal word length containing Zimin patterns, advancing the quantitative understanding of unavoidable patterns in words.

## Key findings

- Established tight bounds for the function f(n,q) for Zimin patterns.
- Determined f(3,q) up to a constant factor as Θ(2^q q!).
- Extended the understanding of unavoidable patterns in combinatorics on words.

## Abstract

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 $f(3,q) = \Theta(2^q q!)$.