Does every large enough string have repeats?
No, unfortunately not. There are even infinite square-free words if your alphabet has at least three symbols.

This apparently natural border (two-element alphabets have only finitely many square-free words) is observed in many places, for instance:


$\{xyyz \mid x,y,z\in \Sigma^+\}$ is co-finite for $|\Sigma|\leq 2$ but not context-free for $\Sigma&gt;2$.
The class of languages generated by terminal-free patterns can be learned in the limit if $|\Sigma|&gt;3$ but not so if $|\Sigma|=2$ [Reid2004].