Squarefree extensions of words

A word is squarefree if it does not contain nonempty factors of the form XX. In 1906 Thue proved that there exist arbitrarily long squarefree words over a 3-letter alphabet. It was proved recently that among these words there are infinitely many extremal ones, that is, having a square in every single-letter extension. We study diverse problems concerning extensions of words preserving the property of avoiding squares. Our main motivation is the conjecture stating that there are no extremal words over a 4-letter alphabet. We also investigate a natural recursive procedure of generating squarefree words by a single-letter rightmost extension. We present the results of computer experiments supporting a supposition that this procedure gives an infinite squarefree word over any alphabet of size at least three.