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Let $[3]^n$ be the set of all length $n$ strings over the alphabet $1, 2, 3$. A combinatorial line is a set of three points in $[3]^n$, formed by taking a string with one or more wildcards $x$ in it, e.g., $112x1xx3\ldots$, and replacing those wildcards by $1, 2$ and $3$, respectively. In the example given, the resulting combinatorial line is: $\{ 11211113\ldots, 11221223\ldots, 11231333\ldots \}$. A subset of $[3]^n$ is said to be line-free if it contains no lines. Let $c_n$ be the size of the largest line-free subset of $[3]^n$.
Density Hales-Jewett theorem: $lim_{n \rightarrow \infty} c_n/3^n = 0$