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

Extremal overlap-free and extremal $\beta$-free binary words