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