# Corners

(Redirected from Corner)
A corner is a subset of $[n]^2$ of the form $\{(x,y),(x+d,y),(x,y+d)\}$ with $d\ne 0.$ One often insists also that d should be positive.
The corners theorem asserts that for every $\delta\gt0$ there exists n such that every subset A of $[n]^2$ of density at least $\delta$ contains a corner.
In general, a corner is a subset of $[n]^m$ of the form $\{(x_1,x_2,\ldots , x_m),(x_1+d,x_2,\ldots , x_m),(x_1,x_2+d,\ldots , x_m),\ldots ,(x_1,x_2,\ldots , x_m+d)\}$ with $d\ne 0.$
The Multidimensional Szemeredi's theorem (proved by Furstenberg and Katznelson) asserts that for every real $\delta\gt0$ and integer $m\gt1$ there exists n such that every subset A of $[n]^m$ of density at least $\delta$ contains a corner.