Corners

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

The corners theorem asserts that for every [math]\delta\gt0[/math] there exists n such that every subset A of [math][n]^2[/math] of density at least [math]\delta[/math] contains a corner.

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

The Multidimensional Szemeredi's theorem (proved by Furstenberg and Katznelson) asserts that for every real [math]\delta\gt0[/math] and integer [math]m\gt1[/math] there exists n such that every subset A of [math][n]^m[/math] of density at least [math]\delta[/math] contains a corner.