Unavoidable pattern

In mathematics and theoretical computer science, an unavoidable pattern is a pattern of symbols that must occur in any sufficiently long string over an alphabet. An avoidable pattern is one for which there are infinitely many words no part of which match the pattern.

Zimin words are an example of unavoidable patterns. These are formed by inserting a new letter between two instances of the previous Zimin word, i.e., the first Zimin word A is used to create the second Zimin word ABA, which in turn creates ABACABA and then ABACABADABACABA and so on.

It is unavoidable that any string, containing two unique characters, that is five or more characters long will contain a pattern of the form ABA (the second Zimin word). Using three unique characters any string containing 29 or more characters will contain a pattern of the form ABACABA

Let A be an alphabet of letters and E a disjoint alphabet of pattern symbols or "variables". Elements of E⁺ are patterns. For a pattern p, the pattern language is that subset of A^∗ containing all words h(p) where h is a non-erasing semigroup morphism from the free monoid E^∗ to A^∗. A word w in A^∗ matches or meets p if it contains some word in the pattern language as a factor, otherwise w avoids p.

A pattern p is avoidable on A if there are infinitely many words in A^∗ that avoid p; it is unavoidable on A if all sufficiently long words in A^∗ match p. We say that p is k-unavoidable if it is unavoidable on every alphabet of size k and correspondingly k-avoidable if it is avoidable on an alphabet of size k.

There is a word W(k) over an alphabet of size 4k which avoids every avoidable pattern with less than 2k variables.

The avoidability index of a pattern p is the smallest k such that p is k-avoidable, or ∞ if p is unavoidable. For binary patterns (two variables x and y) we have:

A square-free word is one avoiding the pattern xx. An example is the word over the alphabet {0,±1} obtained by taking the first difference of the Thue–Morse sequence.

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