Corners

A corner is a subset of [math]\displaystyle{ [n]^2 }[/math] of the form [math]\displaystyle{ \{(x,y),(x+d,y),(x,y+d)\} }[/math] with [math]\displaystyle{ d\ne 0. }[/math] One often insists also that d should be positive.

The corners theorem asserts that for every [math]\displaystyle{ \delta\gt 0 }[/math] there exists n such that every subset A of [math]\displaystyle{ [n]^2 }[/math] of density at least [math]\displaystyle{ \delta }[/math] contains a corner.

In general, a corner is a subset of [math]\displaystyle{ [n]^m }[/math] of the form [math]\displaystyle{ \{(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)\} }[/math] with [math]\displaystyle{ d\ne 0. }[/math]

The Multidimensional Szemeredi's theorem (proved by Furstenberg and Katznelson) asserts that for every real [math]\displaystyle{ \delta\gt 0 }[/math] and integer [math]\displaystyle{ m\gt 1 }[/math] there exists n such that every subset A of [math]\displaystyle{ [n]^m }[/math] of density at least [math]\displaystyle{ \delta }[/math] contains a corner.