No extremal square-free words over alphabets of size at least 5

Abstract

A word over an alphabet $\mathbb A$ contains a square if it has a subword of the form $XX$ where $X$ is a word. A word $W$ is \emph{extremal square-free} if it does not contain a square, but it contains a square as soon as any letter of $\mathbb A$ is inserted at any position of $W$. Grytczuk, Kordulewski, and Niewiadomski conjectured that there are no extremal square-free words over alphabets of size at least 4. We prove this for alphabets of size at least 5. Our proof also implies that the sequence of \emph{nonchalant words} defined by Grytczuk, Kordulewski, and Niewiadomski is infinite and converges to an infinite word for all alphabets of size at least 5.