Extremal overlap-free and extremal $��$-free binary words

An overlap-free (or $��$-free) word $w$ over a fixed alphabet $��$ is extremal if every word obtained from $w$ by inserting a single letter from $��$ at any position contains an overlap (or a factor of exponent at least $��$, respectively). We find all lengths which admit an extremal overlap-free binary word. For every extended real number $��$ such that $2^+\leq��\leq 8/3$, we show that there are arbitrarily long extremal $��$-free binary words.