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and notationsŸ¨ óŸ¨Definitions and notationsŸ¨Recall:óŸ¨Definitions and notationsŸ¨ óŸ¨The Van der Waerden conjectureª, Ÿ¨ ª óŸ¨ PreparationsŸ¨'Theorem 12.1: Proof: By Theorem 5.4¡$( €óŸ¨ Preparationsª Ÿ¨Def:ó Ÿ¨ Preparationsª Ÿ¨Note:ó Ÿ¨ Preparationsª Ÿ¨Fact:ó  Ÿ¨ Preparationsª Ÿ¨ Theorem 12.2:ó$ Ÿ¨Proof of Theorem 12.2ªŸ¨ ó' Ÿ¨Proof of Theorem 12.2ªŸ¨ ª ó)Ÿ¨Proof of Theorem 12.2ªŸ¨ ª ó+Ÿ¨Proof of Theorem 12.2ªŸ¨ ª ó-Ÿ¨Proof of Theorem 12.2ªŸ¨ ª ó/Ÿ¨Proof of Theorem 12.2ªŸ¨ ª ó0Ÿ¨ Preparationsª Ÿ¨ Problem 12A:ó3Ÿ¨ Preparationsª Ÿ¨Corollary (i) of Theorem 12.2:ó5Ÿ¨ Preparationsª Ÿ¨LCorollary (ii) of Theorem 12.2: Proof: By Theorem 12.2 and Problem 12A. ¡"LZZMª *ó7Ÿ¨ Preparationsª Ÿ¨ Problem 12B:ó9Ÿ¨ Preparationsª Ÿ¨MTheorem 12.3: Proof: By Larange Multiplier, Theorem 12.2, and Problem 12B.ª,ó;Ÿ¨ Preparationsª Ÿ¨KTheorem 12.4: Proof: By Corollary (ii) of Theorem 12.2 and Theorem 12.3.ó=Ÿ¨ Preparationsª Ÿ¨Def:ó?Ÿ¨ Preparationsª Ÿ¨ Theorem 12.5:óAŸ¨ Preparationsª Ÿ¨ óCŸ¨ Preparationsª 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+îåï € üðô ðìðŒð( ð ðìðr ðì S ð€¤rt ¿ÿ˜ð`°˜Pðà  t  ð žðr ðì S ð€€Ðt ¿ÿ˜ð€° € ðà t  ð žð`² ðì c ð$A N?¿ÿ?ðŠ¿ z ð Á NðH ðì ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿÿ¸ÿÿïf²p=ÿˆ‘Љº___PPT10‹ië.A ÆP�ÔÑ+Dñ='ñ ¹ÿÿÿÿ=ñ @Bñ +îåï € üðô°ðððŒð( ð ðððr ðð S ð€x›t ¿ÿ˜ð`°˜Pðà  t  ð žðr ðð S ð€Út ¿ÿ˜ð€° € ðà t  ð žð`² ðð c ð$A% V?¿ÿ?ðŠ¿9Ó ð Á VðH ðð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿÿ¸ÿÿïf²p=ÿˆ‘Љº___PPT10‹ië.C Æ@›™¼+Dñ='ñ ¹ÿÿÿÿ=ñ @Bñ +îåï € üðôÀðôðŒð( ð ðôðr ðô S ð€LÑt ¿ÿ˜ð`°˜Pðà  t  ð žðr ðô S ð€´…t ¿ÿ˜ð€° € ðà t  ð žð`² ðô c ð$A S?¿ÿ?ðŠ¿Á0 ð Á SðH ðô ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿÿ¸ÿÿïf²p=ÿˆ‘Љº___PPT10‹ië.F Æ�6+Dñ='ñ ¹ÿÿÿÿ=ñ @Bñ +îåï € üðôÐðøðŒð( ð ðøðr ðø S ð€üt ¿ÿ˜ð`°˜Pðà  t  ð žðr ðø S ð€œ7Æ ¿ÿ˜ð€° € ðà Æ  ð žð`² ðø c ð$A Z?¿ÿ?ðŠ¿¤• ð Á ZðH ðø ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿÿ¸ÿÿïf²p=ÿˆ‘Љº___PPT10‹ië.H ÆÀq§g+Dñ='ñ ¹ÿÿÿÿ=ñ @Bñ +îåï € üðôàðüðŒð( ð ðüðr ðü S ð€$Ët ¿ÿ˜ð`°˜Pðà  t  ð žðr ðü S ð€FÆ ¿ÿ˜ð€° € ðà Æ  ð žð`² ðü c ð$A [?¿ÿ?ðŠ¿ï ð Á [ðH ðü ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿÿ¸ÿÿïf²p=ÿˆ‘Љº___PPT10‹ië.J Æ sÀ$+Dñ='ñ ¹ÿÿÿÿ=ñ @Bñ +îåï € üðôðððŒð( ð ððr ð S ð€|&t ¿ÿ˜ð`°˜Pðà  t  ð žðr ð S ð€€#Æ ¿ÿ˜ð€° € ðà Æ  ð žð`² ð c ð$A \?¿ÿ?ð"‘& ð Á \ðH ð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿÿ¸ÿÿïf²p=ÿˆ‘Љº___PPT10‹ië.J Æ_•Ú+Dñ='ñ ¹ÿÿÿÿ=ñ @Bñ +îåï € üðôððŒð( ð ððr ð S ð€ ×t ¿ÿ˜ð`°˜Pðà  t  ð žðr ð S ð€ IÆ ¿ÿ˜ð€° € ðà Æ  ð žð`² ð c ð$A ]?¿ÿ?ðPd÷ ð Á ]ðH ð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿÿ¸ÿÿïf²p=ÿˆ‘Љº___PPT10‹ië.L ÆðÆÒ&+Dñ='ñ ¹ÿÿÿÿ=ñ @Bñ +îåï € üðôððŒð( ð ððr ð S ð€ì&Æ ¿ÿ˜ð`°˜Pðà  Æ  ð žðr ð S ð€ˆ»î ¿ÿ˜ð€° € ðà î  ð žð`² ð c ð$A `?¿ÿ?ðõ‘Vë ð Á `ðH ð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿÿ¸ÿÿïf²p=ÿˆ‘Љº___PPT10‹ië.L Æ0©ƒß+Dñ='ñ ¹ÿÿÿÿ=ñ @Bñ +îåï € üðô ð ðŒð( ð ð ðr ð  S ð€\t ¿ÿ˜ð`°˜Pðà  t  ð žðr ð  S ð€Ô Æ ¿ÿ˜ð€° € ðà Æ  ð žð`² ð  c ð$A a?¿ÿ?ðÈ¿¡æ ð Á aðH ð  ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿÿ¸ÿÿïf²p=ÿˆ‘Љº___PPT10‹ië.N ƀå—+Dñ='ñ ¹ÿÿÿÿ=ñ @Bñ +îåï € üðô0ððŒð( ð ððr ð S ð€ÜaÆ ¿ÿ˜ð`°˜Pðà  t  ð žðr ð S ð€$eÆ ¿ÿ˜ð€° € ðà Æ  ð žð`² ð c ð$A b?¿ÿ?ð"¿±³ ð Á bðH ð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿÿ¸ÿÿïf²p=ÿˆ‘Љº___PPT10‹ië.O Æ�ð3�+Dñ='ñ ¹ÿÿÿÿ=ñ @Bñ +îåï € üðô@ððŒð( ð ððr ð S ð€p�Æ ¿ÿ˜ð`°˜Pðà  Æ  ð žðr ð S ð€ÛÆ ¿ÿ˜ð€° € ðà Æ  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s¹ëíX·!ÏÍM`ÿl—„2ð$²45ð@7áÿáó´ïë“$ÿ†2ð$¨4*€¡\ Iyþ¸`B‰ÿr ‰2ð$?°1'©ö{ç@q‡(i+jÿ´’�ƒ ð0�ƒ†A¿ÀÅAÿ@ñ÷ð8ó€ó €ÐwçˆR Êš;¼;k8Êš;úgþý4BdBdP¥z[ 0Žÿÿÿÿÿÿpûppû@ <ý4dddd|¥ß 0¸¦Ð0ý<ý4!d!d|¥ß 0¸¦Ð0ýÿ ˆ8Š0º___PPT10‹ pµpµ?Ù Ú %ðPóŸ¨The Van der Waerden conjecture¡6ª, Ÿ speaker:YO W‰[¡ Z  ªóŸ¨ContentsŸ¨oDefinitions and notations The Van der Waerden conjecture Preparations A proof of the Van der Waerden conjectureªP", ó Ÿ¨Definitions and notationsŸ¨ óŸ¨Definitions and notationsŸ¨Recall:óŸ¨Definitions and notationsŸ¨ óŸ¨The Van der Waerden conjectureª, Ÿ¨ ª óŸ¨ PreparationsŸ¨'Theorem 12.1: Proof: By Theorem 5.4¡$( €óŸ¨ PreparationsŸ¨Def:ó Ÿ¨ PreparationsŸ¨Note:ó Ÿ¨ PreparationsŸ¨Fact:ó  Ÿ¨ PreparationsŸ¨ Theorem 12.2:ó$ Ÿ¨Proof of Theorem 12.2Ÿ¨ ó' Ÿ¨Proof of Theorem 12.2Ÿ¨ ª ó)Ÿ¨Proof of Theorem 12.2Ÿ¨ ª ó+Ÿ¨Proof of Theorem 12.2Ÿ¨ ª ó-Ÿ¨Proof of Theorem 12.2Ÿ¨ ª ó/Ÿ¨Proof of Theorem 12.2Ÿ¨ ª ó0Ÿ¨ PreparationsŸ¨ Problem 12A:ó3Ÿ¨ PreparationsŸ¨Corollary (i) of Theorem 12.2:ó5Ÿ¨ PreparationsŸ¨LCorollary (ii) of Theorem 12.2: Proof: By Theorem 12.2 and Problem 12A. ¡"LZZMó7Ÿ¨ PreparationsŸ¨ Problem 12B:ó9Ÿ¨ PreparationsŸ¨NTheorem 12.3: Proof: By Lagrange Multiplier, Theorem 12.2, and Problem 12B.ó;Ÿ¨ PreparationsŸ¨KTheorem 12.4: Proof: By Corollary (ii) of Theorem 12.2 and Theorem 12.3.ó=Ÿ¨ PreparationsŸ¨Def:ó?Ÿ¨ PreparationsŸ¨ Theorem 12.5:óAŸ¨ PreparationsŸ¨ óCŸ¨ PreparationsŸ¨7Theorem 12.6: Proof: By induction and Theorem 12.5.óEŸ¨ PreparationsŸ¨Corollary of Theorem 12.6:óGŸ¨ PreparationsŸ¨ óIŸ¨ PreparationsŸ¨FTheorem 12.7: Proof: By Theorem 12.4 and Corollary of Theorem 12.6.óKŸ¨)A proof of the Van der Waerden conjectureª, Ÿ¨Substitution Principle:óQ Ÿ¨)A proof of the Van der Waerden conjectureª, Ÿ¨ ª óS!Ÿ¨)A proof of the Van der Waerden conjectureª, Ÿ¨ ª óV"Ÿ¨)A proof of the Van der Waerden conjectureª, Ÿ¨ ª óW#Ÿ¨)A proof of the Van der Waerden conjectureª, Ÿ¨ óY$Ÿ¨ Ÿ Thank all ^_^& ¡Hêrûxõ#? ×x±YÇ