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

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