DHJ(3)

The k=3 Density Hales-Jewett theorem (DHJ(3)) has many equivalent forms. Here is one:

DHJ(3), Version 1. For every [math]\displaystyle{ \delta \gt 0 }[/math] there exists n such that every subset [math]\displaystyle{ A \subset [3]^n }[/math] of density at least [math]\displaystyle{ \delta }[/math] contains a combinatorial line.

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 subset of [math]\displaystyle{ [3]^n }[/math] is said to be line-free if it contains no combinatorial lines. Let [math]\displaystyle{ c_n }[/math] be the size of the largest line-free subset of [math]\displaystyle{ [3]^n }[/math].

DHJ(3), Version 2 [math]\displaystyle{ \lim_{n \rightarrow \infty} c_n/3^n = 0 }[/math].