Corners theorem

Statement

In arithmetic combinatorics, the corners theorem states that for every $\varepsilon>0$, for large enough $N$, any set of at least $\varepsilon N^2$ points in the $N\times N$ grid $\{1,\ldots,N\}^2$ contains a corner, i.e., a triple of points of the form $\{(x,y), (x+h,y), (x,y+h)\}$ with $h\ne 0$. It was first proved by Miklós Ajtai and Endre Szemerédi in 1974 using Szemerédi's theorem. In 2003, József Solymosi gave a short proof using the triangle removal lemma.