# 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^+≤β≤ 8/3, we show that there are arbitrarily long extremal β-free binary words.

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