No extremal square-free words over large alphabets

TL;DR

This paper investigates the existence of extremal square-free words across different alphabet sizes, proving their infinite existence for three-letter alphabets and non-existence for alphabets of size 17 or more.

Contribution

It establishes the non-existence of extremal square-free words over large alphabets (size ≥17), extending previous results on their existence over smaller alphabets.

Findings

Existence of infinitely many ternary extremal square-free words.

Non-existence of extremal square-free words over alphabets of size at least 17.

Abstract

A word is square-free if it does not contain any square (a word of the form $XX$), and is extremal square-free if it cannot be extended to a new square-free word by inserting a single letter at any position. Grytczuk, Kordulewski, and Niewiadomski proved that there exist infinitely many ternary extremal square-free words. We establish that there are no extremal square-free words over any alphabet of size at least 17.