A Note on Antichains of Words

We can compress the word `banana' as xyyz, where x = `b', y = `an',z = `a'. We say that `banana' encounters yy. Thus a `coded' version of yy shows up in `banana'. The relation `u encounters w' is transitive, and thus generates an order on words. We study antichains under this order. In particular we show that in this order there is an infinite antichain of binary words avoiding overlaps. AMS Subject Classification: 68R15, 06A99 Key Words: overlaps. antichains, words avoiding patterns 1 Introduction The study of words avoiding patterns is an area of combinatorics on words reaching back at least to the turn of the century, when Thue proved [29] that one can find arbitrarily long words over a 3 letter alphabet in which no two adjacent subwords are identical. If w is such a word, then w cannot be written w = xyyz with y a non-empty word. In modern parlance, we would say that w avoids yy. A word which can be written as xyyz is said to encounter yy. Thue also showed that there are ar..