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1、MartinAignerGiinterM.ZieglerProofsfromTHEBOOKThirdEditionWith250FiguresIncludingIllustrationsbyKarlH.HofmannSpringerPrefacePaulErd˝oslikedtotalkaboutTheBook,inwhichGodmaintainstheperfectproofsformathematicaltheorems,followingthedictumofG.H.Hardythatthereisnop
2、ermanentplaceforuglymathematics.Erd˝osalsosaidthatyouneednotbelieveinGodbut,asamathematician,youshouldbelieveinTheBook.Afewyearsago,wesuggestedtohimtowriteupafirst(andverymodest)approximationtoTheBook.Hewasenthusiasticabouttheideaand,characteristically,wenttow
3、orkimmediately,fillingpageafterpagewithhissuggestions.OurbookwassupposedtoappearinMarch1998asapresenttoErd˝os'85thbirthday.WithPaul'sunfortunatedeathinthesummerof1997,heisnotlistedasaco-author.Insteadthisbookisdedicatedtohismemory.PaulErd˝osWehavenodefinitionor
4、characterizationofwhatconstitutesaprooffromTheBook:allweofferhereistheexamplesthatwehaveselected,hop-ingthatourreaderswillshareourenthusiasmaboutbrilliantideas,cleverinsightsandwonderfulobservations.Wealsohopethatourreaderswillenjoythisdespitetheimperfections
5、ofourexposition.TheselectionistoagreatextentinfluencedbyPaulErd˝oshimself.Alargenumberofthetopicsweresuggestedbyhim,andmanyoftheproofstracedirectlybacktohim,orwereinitiatedbyhissupremeinsightinaskingtherightquestionorinmakingtherightconjecture.Sotoalargeextent
6、thisbookreflectstheviewsofPaulErd˝osastowhatshouldbeconsideredaprooffromTheBook.“TheBook”Alimitingfactorforourselectionoftopicswasthateverythinginthisbookissupposedtobeaccessibletoreaderswhosebackgroundsincludeonlyamodestamountoftechniquefromundergraduatemathe
7、matics.Alittlelinearalgebra,somebasicanalysisandnumbertheory,andahealthydollopofelementaryconceptsandreasoningsfromdiscretemathematicsshouldbesufficienttounderstandandenjoyeverythinginthisbook.Weareextremelygratefultothemanypeoplewhohelpedandsupporteduswiththi
8、sproject—amongthemthestudentsofaseminarwherewediscussedapreliminaryversion,toBennoArtmann,StephanBrandt,StefanFelsner,EliGoodman,TorstenHeldmann,andHansMielke.WethankMargritBarrett,Christ
