Bounds on Zimin Word Avoidance

TL;DR

This paper investigates the maximum length of words that can avoid certain unavoidable patterns called Zimin words, providing bounds on their avoidability in combinatorial word theory.

Contribution

It introduces bounds on the length of words that can avoid Zimin words, advancing understanding of pattern avoidance in combinatorics on words.

Findings

Established upper bounds on word length avoiding Zimin words

Characterized the structure of avoidable words for Zimin patterns

Extended previous results on pattern avoidance in finite alphabets

Abstract

How long can a word be that avoids the unavoidable? Word $W$ encounters word $V$ provided there is a homomorphism $\phi$ defined by mapping letters to nonempty words such that $\phi(V)$ is a subword of $W$. Otherwise, $W$ is said to avoid $V$. If, on any arbitrary finite alphabet, there are finitely many words that avoid $V$, then we say $V$ is unavoidable. Zimin (1982) proved that every unavoidable word is encountered by some word $Z_n$, defined by: $Z_1 = x_1$ and $Z_{n+1} = Z_n x_{n+1} Z_n$. Here we explore bounds on how long words can be and still avoid the unavoidable Zimin words.