Corners theorem: Difference between revisions

Revision as of 06:54, 6 July 2009

@@ Line 1: / Line 1: @@
-'''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>.
+goway tours .japan http://annabuch71.tripod.com/dr.-drama-canon.html dr. drama canonwhos afraid of http://annabuch71.tripod.com/1.50-update-japan.html 1.50 update japandj drama a.i http://annabuch71.tripod.com/j.moss-afraid.html j.moss afraidsoundtrack who s afraid of http://annabuch71.tripod.com/2006-drama-series-te.html 2006 drama series temr. goodbye korean drama http://annabuch71.tripod.com/info-ssi-coordination.fr.html info ssi coordination.frmy exwifee .com http://annabuch71.tripod.com/drama-ministry.com.html drama ministry.comceleb drama .com http://annabuch71.tripod.com/u.s-embassy-in-tokyo-japan.html u.s embassy in tokyo japankoh i noor t.v. drama http://annabuch71.tripod.com/allhotgirls.com-raven-riley.html allhotgirls.com raven rileynot afraid.mp3 http://annabuch71.tripod.com/jk-japan-co.html jk japan cosela ward t.v. drama http://annabuch71.tripod.com/tomato-canned-.edu.html tomato canned .edu
-'''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>.
-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)]].
-One consequence of the corners theorem is that any subset of the triangular grid
-:<math>\Delta_n = \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c=n\}</math>
-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>.
-A [[DHJ(1,3)|special case of the corners theorem]] is of interest in connection with [[DJH(1,3)]].

Corners theorem: Difference between revisions

Revision as of 06:54, 6 July 2009

Navigation menu

Page actions

Page actions

Personal tools

Navigation

Search

Tools