Sequence avoiding any complete word (Q1123274)

A word u is said to avoid a word v is the subwords of the word u there are no instances of the word v. We recall that an instance of the word v is the result of substituting certain words for letters of the word v; also identical letters are replaced by identical words. We will say that a sequence of words \(\{u_ i|\) \(i<\omega \}\) avoids the word v if each word u, avoids v. We call a word complete if it does not contain any letters which occur exactly once, and for any different letters, x, y of this word, the words xy and yx are subwords. In this note we prove the following Theorem: There exists a sequence of words in an alphabet on four letters which avoids every complete word.