Polynomial versus exponential growth in repetition-free binary words

It is known that the number of overlap-free binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 73. More precisely, there are only polynomially many binary words of length n that avoid 73 -powers, but there are exponentially many binary words of length n that avoid 73+-powers. This answers an open question of Kobayashi from 1986.