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The [[corners theorem]] asserts that for every <math>\delta>0</math> there exists n such that every subset A of <math>[n]^2</math> of density at least <math>\delta</math> contains a corner. | The [[corners theorem]] asserts that for every <math>\delta>0</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>0</math> and integer <math>m>1</math> there exists n such that every subset A of <math>[n]^m</math> of density at least <math>\delta</math> contains a corner. | |||
Latest revision as of 17:17, 8 March 2009
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.
