Extremal Square-free Words

TL;DR

This paper introduces the concept of extremal square-free words, proving their infinite existence over a 3-letter alphabet and utilizing computer verification for parts of the construction.

Contribution

It defines extremal square-free words and demonstrates their infinite existence over a 3-letter alphabet, advancing understanding of combinatorial word structures.

Findings

Existence of infinitely many extremal square-free words over 3-letter alphabet

Construction relies on computer verification

Poses open problems related to extremal words

Abstract

A word is \emph{square-free} if it does not contain non-empty factors of the form $XX$. In 1906 Thue proved that there exist arbitrarily long square-free words over $3$-letter alphabet. We consider a new type of square-free words. A square-free word is \emph{extremal} if it cannot be extended to a new square-free word by inserting a single letter on arbitrary position. We prove that there exist infinitely many extremal words over $3$-letter alphabet. Some parts of our construction relies on computer verifications. We also pose some related open problems.