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Erschienen in:
14.03.2019
On Long Words Avoiding Zimin Patterns
Erschienen in: Theory of Computing Systems | Ausgabe 5/2019
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Abstract
A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern p is unavoidable if, over every finite alphabet, every sufficiently long word encounters p. A theorem by Zimin and independently by Bean, Ehrenfeucht and McNulty states that a pattern over n distinct variables is unavoidable if, and only if, p itself is encountered in the n-th Zimin pattern. Given an alphabet size k, we study the minimal length f(n,k) such that every word of length f(n,k) encounters the n-th Zimin pattern. It is known that f is upper-bounded by a tower of exponentials. Our main result states that f(n,k) is lower-bounded by a tower of n − 3 exponentials, even for k = 2. To the best of our knowledge, this improves upon a previously best-known doubly-exponential lower bound. As a further result, we prove a doubly-exponential upper bound for encountering Zimin patterns in the abelian sense.