Roth's theorem

Roth's theorem ([math]\displaystyle{ {\Bbb Z}/N{\Bbb Z} }[/math] version) If N is sufficiently large depending on [math]\displaystyle{ \delta \gt 0 }[/math], then any subset A of [math]\displaystyle{ [N] }[/math] of density at least [math]\displaystyle{ \delta }[/math] contains an arithmetic progression x, x+r, x+2r with [math]\displaystyle{ r \gt 0 }[/math].

Roth's theorem ([math]\displaystyle{ [3]^n }[/math] version) If n is sufficiently large depending on [math]\displaystyle{ \delta \gt 0 }[/math], then any subset of [math]\displaystyle{ [3]^n }[/math] of density at least [math]\displaystyle{ \delta }[/math] contains an algebraic line, i.e. a triple (x,x+r,x+2r) where r is non-zero and we identify [math]\displaystyle{ [3]^n }[/math] with [math]\displaystyle{ ({\Bbb Z}/3{\Bbb Z})^n }[/math].

Roth's theorem is a special case of Szemerédi's theorem. See also Szemerédi's combinatorial proof of Roth's theorem.

Roth's theorem is implied by the corners theorem, which in turn is implied by the k=3 case of the IP-Szemerédi theorem, which is in turn implied by k=3 case of the density Hales-Jewett theorem.