# Corners theorem

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

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

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

$\Delta_n = \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c=n\}$

of density at least $\delta$ will contain an equilateral triangle $(a+r,b,c),(a,b+r,c),(a,b,c+r)$ with $r\gt0$, if n is sufficiently large depending on $\delta$.

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