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| '''Corners theorem''': (<math>{\Bbb Z}/N{\Bbb Z}</math> version) If N is sufficiently large depending on <math>\delta</math>, then any <math>\delta</math>-dense subset of <math>{}[N]^2</math> must contain a "corner" (x,y), (x+r,y), (x,y+r) with <math>r > 0</math>.
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| '''Corners theorem''': (<math>({\Bbb Z}/3{\Bbb Z})^n</math> version) If n is sufficiently large depending on <math>\delta</math>, then any <math>\delta</math>-dense subset of <math>{}(({\Bbb Z}/3{\Bbb Z})^n)^2</math> must contain a "corner" (x,y), (x+r,y), (x,y+r) with <math>r \neq 0</math>.
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| This result was first proven by [[Ajtai-Szemerédi's proof of the corners theorem|Ajtai and Szemerédi]]. A simpler proof, based on the [[triangle removal lemma]], was obtained by Solymosi. The corners theorem implies [[Roth's theorem]] and is in turn implied by the [[IP-Szemerédi theorem]], which in turn follows from [[DHJ(3)]].
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| One consequence of the corners theorem is that any subset of the triangular grid
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| :<math>\Delta_n = \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c=n\}</math>
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| of density at least <math>\delta</math> will contain an equilateral triangle <math>(a+r,b,c),(a,b+r,c),(a,b,c+r)</math> with <math>r>0</math>, if n is sufficiently large depending on <math>\delta</math>.
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| A [[DHJ(1,3)|special case of the corners theorem]] is of interest in connection with [[DJH(1,3)]].
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