# square-free word

I am trying to understand one concept. Is it possible to have a square-free(no subwords ss) word in a language of just $\{0,1\}$ of any given length ( $10^{100}$ for instance). I found out a Thue–Morse sequence , but it is still not really the thing. If there is no such word , how to prove it?

• yes consider $01001000100001....$ (once zero, one, twice zero, one, triple zero...) Sep 14, 2018 at 16:31
• @Yanko but , isn't 00 a subword? Sep 14, 2018 at 16:33
• It is, what is your definition for square-free? I understand it as "a word $w$ is square free if there is no sub-word $s$ such that $w=ss$" Is that what you mean? Sep 14, 2018 at 16:35
• Is there a square-free word of lenth $4$? If there is no square-free word of length $4$, how can there be a square-free word of length $10^{100}$?
– bof
Sep 14, 2018 at 17:31
• But that has $00$. Sep 14, 2018 at 17:55

• For an alphabet of $3$ symbols, there are infinitely many square-free words — equivalently, there is an infinitely long square-free word.