Corners theorem

Corners theorem: ([math]\displaystyle{ {\Bbb Z}/N{\Bbb Z} }[/math] version) If N is sufficiently large depending on [math]\displaystyle{ \delta }[/math], then any [math]\displaystyle{ \delta }[/math]-dense subset of [math]\displaystyle{ {}[N]^2 }[/math] must contain a "corner" (x,y), (x+r,y), (x,y+r) with [math]\displaystyle{ r \gt 0 }[/math].

Corners theorem: ([math]\displaystyle{ ({\Bbb Z}/3{\Bbb Z})^n }[/math] version) If n is sufficiently large depending on [math]\displaystyle{ \delta }[/math], then any [math]\displaystyle{ \delta }[/math]-dense subset of [math]\displaystyle{ {}(({\Bbb Z}/3{\Bbb Z})^n)^2 }[/math] must contain a "corner" (x,y), (x+r,y), (x,y+r) with [math]\displaystyle{ r \neq 0 }[/math].

This result was first proven by Ajtai and Szemerédi. A simpler proof, based on the triangle removal lemma, was obtained by Solymosi. The corners theorem implies Roth's theorem and is in turn implied by the IP-Szemerédi theorem, which in turn follows from DHJ(3).

One consequence of the corners theorem is that any subset of the triangular grid

[math]\displaystyle{ \Delta_n = \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c=n\} }[/math]

of density at least [math]\displaystyle{ \delta }[/math] will contain an equilateral triangle [math]\displaystyle{ (a+r,b,c),(a,b+r,c),(a,b,c+r) }[/math] with [math]\displaystyle{ r\gt 0 }[/math], if n is sufficiently large depending on [math]\displaystyle{ \delta }[/math].

A special case of the corners theorem is of interest in connection with DHJ(1,3).