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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 in it, e.g., $112*1**3\ldots$, and replacing those wildcards by $1, 2$ and $3$, respectively. In the example given, the resulting combinatorial line is: $$\{ 11211113\ldots, 11221223\ldots, 11231333\ldots \}$$ The Density Hales-Jewett theorem asserts that for any $\delta > 0$, for sufficiently large $n = n(\delta)$, all subsets of $[3]^n$ of size at least $\delta 3^n$ contain a combinatorial line,