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?
2026-03-25 09:33:15.1774431195
square-free word
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The longest square-free word for a language with a binary alphabet is of length 3. However, for alphabets of at least three symbols, there are infinitely many square-free words.
It's easy to prove the first assertion simply by enumerating all binary sequences of length four and observing that all of them contain a square. Since every word of length greater than four contains a subword of length 4, it must also contain a square (the one in the subword, at least).