# Corners theorem

### From Polymath1Wiki

**Corners theorem**: ( version) If N is sufficiently large depending on δ, then any δ-dense subset of [*N*]^{2} must contain a "corner" (x,y), (x+r,y), (x,y+r) with *r* > 0.

**Corners theorem**: ( version) If n is sufficiently large depending on δ, then any δ-dense subset of must contain a "corner" (x,y), (x+r,y), (x,y+r) with .

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

of density at least δ will contain an equilateral triangle (*a* + *r*,*b*,*c*),(*a*,*b* + *r*,*c*),(*a*,*b*,*c* + *r*) with *r* > 0, if n is sufficiently large depending on δ.

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