In combinatorics, a square-free word is a word (a sequence of symbols) that does not contain any squares. A square is a word of the form XX, where X is not empty. Thus, a square-free word can also be defined as a word that avoids the pattern XX.

Over a binary alphabet ${\displaystyle \{0,1\}}$, the only square-free words are the empty word ${\displaystyle \epsilon ,0,1,01,10,010}$, and ${\displaystyle 101}$.

Over a ternary alphabet ${\displaystyle \{0,1,2\}}$, there are infinitely many square-free words. It is possible to count the number ${\displaystyle c(n)}$ of ternary square-free words of length n.

This number is bounded by ${\displaystyle c(n)=\Theta (\alpha ^{n})}$, where ${\textstyle 1.3017597<\alpha <1.3017619}$. The upper bound on ${\displaystyle \alpha }$ can be found via Fekete's Lemma and approximation by automata. The lower bound can be found by finding a substitution that preserves square-freeness.

Since there are infinitely many square-free words over three-letter alphabets, this implies there are also infinitely many square-free words over an alphabet with more than three letters.

The following table shows the exact growth rate of the k-ary square-free words, rounded off to 7 digits after the decimal point, for k in the range from 4 to 15:

Consider a map ${\displaystyle {\textbf {w}}}$ from ${\displaystyle \mathbb {N} ^{2}}$ to A, where A is an alphabet and ${\displaystyle {\textbf {w}}}$ is called a 2-dimensional word. Let ${\displaystyle w_{m,n}}$ be the entry ${\displaystyle {\textbf {w}}(m,n)}$. A word ${\displaystyle {\textbf {x}}}$ is a line of ${\displaystyle {\textbf {w}}}$ if there exists ${\displaystyle i_{1},i_{2},j_{1},j_{2}}$such that ${\displaystyle {\text{gcd}}(j_{1},j_{2})=1}$, and for ${\displaystyle t\geq 0,x_{t}=w_{{i_{1}}+{j_{1}t},{i_{2}}+{j_{2}t}}}$.

Carpi proves that there exists a 2-dimensional word ${\displaystyle {\textbf {w}}}$ over a 16-letter alphabet such that every line of ${\displaystyle {\textbf {w}}}$ is square-free. A computer search shows that there are no 2-dimensional words ${\displaystyle {\textbf {w}}}$over a 7-letter alphabet, such that every line of ${\displaystyle {\textbf {w}}}$ is square-free.