Extremal Square-free Words

A word is square-free if it does not contain nonempty factors of the form XX. In 1906 Thue proved that there exist arbitrarily long square-free words over a 3-letter alphabet. We consider a new type of square-free words with an additional property. A square-free word is called extremal if it cannot be extended to a new square-free word by inserting a single letter at any position. We prove that there exist infinitely many square-free extremal words over a 3-letter alphabet. Some parts of our construction relies on computer verifications. It is not known if there exist any extremal square-free words over a 4-letter alphabet.