Extremal Overlap-Free And Extremal Beta-Free Binary Words

An overlap-free (or beta-free) word w over a fixed alphabet Sigma is extremal 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 extremal overlap-free binary word. For every "extended" real number beta such that 2(+) <= beta <= 8/3, we show that there are arbitrarily long extremal beta-free binary words.