Corners theorem
In arithmetic combinatorics, the corners theorem states that for every ε > 0 {\displaystyle \varepsilon >0} , for large enough N {\displaystyle N} , any set of at least ε N 2 {\displaystyle \varepsilon N^{2}} points in the N × N {\displaystyle N\times N} grid { 1 , … , N } 2 {\displaystyle \{1,\ldots ,N\}^{2}} contains a corner, i.e., a triple of points of the form { ( x , y ) , ( x + h , y ) , ( x , y + h ) } {\displaystyle \{(x,y),(x+h,y),(x,y+h)\}} with h ≠ 0 {\displaystyle h\neq 0} . It was first proved by Miklós Ajtai and Endre Szemerédi in 1974 using Szemerédi's theorem.