---
tag: JTKSQGJMJ4
kind: statement
role: proposition
name: Corners theorem
sources:
  - wikipedia: en:31591175@r1365502461
    locator: Corners theorem
    note: adopted
    mention: wikipedia://en/31591175
    sections: [Statement]
same_as:
  wikidata: Q5171652
  wikipedia: Corners theorem
inventory: [N6S8FR2EYF]
---

## 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.