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A pattern of symbols that must occur in any sufficiently long [[word]] over a given alphabet.  An '''avoidable pattern''' is one which for which there are infinitely many words no part of which match the pattern.

Let $A$ be an alphabet of letters and $E$ a disjoint alphabet of pattern symbols or "variables".  Elements of the [[Free semi-group|free semigroup]] of non-empty words $E^+$ are ''patterns''.  For a pattern $p$, the pattern language is that subset of the [[free monoid]] $A^*$ containing all words $h(p)$ where $h$ is a non-erasing [[semigroup morphism]] from  $E^+$ 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 $k$-unavoidable 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 fewer than $2k$ variables.

====Examples====
* The [[Thue–Morse sequence]] avoids the patterns $xxx$ and $xyxyx$.
* The patterns $x$ and $xyx$ are unavoidable on any alphabet.
* The power pattern $xx$ is 3-avoidable; words avoiding this pattern are ''square-free''.
* The power patterns $x^n$ for $n \ge 3$ are 2-avoidable: the Thue–Morse sequence is an example for $n=3$.
* [[Sesquipower]]s are unavoidable.

====Avoidability index====
The '''avoidability index''' of a pattern $p$ is the smallest $k$ such that $p$ is $k$-avoidable, or $\infty$ if $p$ is unavoidable.  For binary patterns (two variables $x$ and $y$) we have:
* $1,x,xy,xyx$ are unavoidable;
* $xx,xxy,xyy,xxyx,xxyy,xyxx,xyxy,xyyx,xxyxx,xxyxy,xyxyy$ have avoidability index 3;
* all other patterns have avoidability index 2.

====Square-free words====

A '''[[square-free word]]''' is one avoiding the pattern $xx$.  An example is the word over the alphabet $\{0,\pm1\}$ obtained by taking the [[first difference]] of the Thue–Morse sequence.

====References====

* Allouche, Jean-Paul; Shallit, Jeffrey; "Automatic Sequences: Theory, Applications, Generalizations" Cambridge University Press (2003)   ISB: 978-0-521-82332-6 {{ZBL|1086.11015}}
* Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V.; "Combinatorics on words. Christoffel words and repetitions in words", CRM Monograph Series '''27''' American Mathematical Society (2009) ISBN 978-0-8218-4480-9  {{ZBL|1161.68043}}
* Lothaire, M.; "Algebraic combinatorics on words", Encyclopedia of Mathematics and Its Applications '''90'''Cambridge University Press (2011) ISBN 978-0-521-18071-9 {{ZBL|1221.68183}}
* Pytheas Fogg, N.; "Substitutions in dynamics, arithmetics and combinatorics"  Lecture Notes in Mathematics '''1794''' Springer-Verlag (2002) ISBN 3-540-44141-7 {{ZBL|1014.11015}}