Corners theorem

In mathematics, the corners theorem is an important result, proved by Miklós Ajtai and Endre Szemerédi, of a statement in arithmetic combinatorics. It states that for every ε > 0 there exists N such that given at least εN² points in the N × N grid {1, ..., N} × {1, ..., N}, there exists a corner, i.e., three points in the form (x, y), (x + h, y), and (x, y + h). Later, Solymosi (2003) gave a simpler proof, based on the triangle removal lemma. The corners theorem implies Roth's theorem.