Journal of Failed Attempts - Colgate

1 , Z+hasthepropertythatevery2-coloringofZ+a dmitsarbitrarilylongmonochromaticarithme ticprogressionswithcommondi ; :(1)achronicleofmybattlewithwhatIconside raparticularlydi cultconjecture;(2)topresentmyprogressont hisconjecture;and(3) , , ,somuchsothatI ,ifwearetotakeawayonemessagefromSteinbec k sOfMiceandMen,it sthatsometimestherabbitdoesn sUp,Doc?Ramseytheorymaybestbesummedupas thestudyofthepreservationofstructuresund ersetpartitions ,arithmeticprogressions, [grantnumberH98230-10-1-0204].12 AARONROBERTSON[17].Forthisarticle,wewill restrictourattentiontothepositiveinteger s,andourinvestigationtothesetofarithmeti cprogressions(ourstructure).

2 AsiscommoninRamseytheory, ,forr2Z+,anr-coloringofthepositiveintege rsisdefinedby :Z+!{0,1,..,r 1}.WesaythatS Z+ismonochromaticunder if| (S)|= ,and,subsequently,statethe2-LargeConject ure,weturntoafundamentalresultinRamseyth eoryontheintegers:vanderWaerden sTheorem[20].Theorem1(vanderWaerden sTheorem).Foranyfixedpositiveintegerskan dr,everyr-coloringofZ+ ,wecannotbreaktheexistenceofarithmeticpr ogressionsviasetpartitioningsincevanderW aerden tbelieveme,try2-coloringthefirstnineposi tiveintegerswithoutcreatingamonochromati c3-termarithmeticprogression(I llwait).Sonowthatwe reallonboard,thenextattributeofarithmeti cprogressionstotakenoteofisthattheyarecl osedundertranslationanddilation:ifS={a,a +d,a+2d.}

3 ,a+(k 1)d}isak-termarithmeticprogression,andba ndcarepositiveintegers,thenc+bS={(ab+c), (ab+c)+bd,(ab+c)+2bd,..,(ab+c)+(k 1)bd} ordsusasimpleinductiveargumentwhenprovin gvanderWaerden ,assumingthatther=2caseofTheorem1istrue( forallk), ,weneedarestatementofTheorem1, (vanderWaerden sTheoremrestatement).Foranyfixedpositive integerskandr,thereexistsaminimuminteger w(k;r)suchthateveryr-coloringof{1,2,..,w (k;r)} (atleastthenontrivialdirection)isgivenby TheCompactnessPrinciple,which,inthissett ing,couldalsobecalledCantor sDiagonalPrinciple, (k;s)existfors=2,3,..,r 1foranyk2Z+.

4 Letm=w(k;r 1)sothatn=w(m;2) ,anarbitraryr-coloringof{1,2,..,n}.Forea seofexposition,letthecolorsberedandr 1di ,suchapersonwouldconcludethatamonochroma ticm-termarithmeticprogressionexistsunde r .Ifthismonochromaticprogressionisred,wea redone,soweassumethatitis blue. Letitbea+d,a+2d,a+3d,..,a+mdandnotethat, sincewecandistinguishbetweenshadesofblue ,wehavean(r 1) (r 1)-coloringsofT={1,2,..,m}anda+dT={a+d,a +2d,a+3d,..,a+md}.Bythedefinitionofmandb ecausearithmeticprogressionsareclosedund ertranslationanddilation,weseethatT,andh encea+dT,admitsamonochromatick-termarith meticprogression, ,thepreviousparagraphisonlyapartialproof sinceImadethesignificantassumptionthatTh eorem1holdsfortwocolors;however,wecansta tethefollowing:(?)

5 Ifevery2-coloringofZ+admitsarbitrarilylo ngmonochromaticarithmeticprogressions,th en,foranyr2Z+,everyr-coloringofZ+ (?), ,Graham,andLandman[7]investigatedastreng theningofTheorem1byrestrictingthesetofal lowablecommondi (r-large,large,D-ap).LetD Z+andletr2Z+.Werefertoanarithmeticprogre ssiona,a+d,a+2d,..,a+(k 1) +,everyr-coloringofZ+4 AARONROBERTSON admitsamonochromatick-termD-ap, +, ,wewouldrestate(?)as:(?)IfZ+is2-large,th enZ+ ,duetoBrown,Graham,andLandman[7],scrawle donit:Conjecture(2-LargeConjecture).LetD Z+.IfDis2-large, :mZ+foranypositiveintegerm(inparticular, thesetofevenpositiveintegers);therangeof anyinteger-valuedpolynomialp(x)withp(0)= 0;anyset{b nc:n2Z+}with ,youmaythinkyouhavespottedtherabbit, , ,whatmakesthisconjecturesoappealing?

6 Firstly,the2-LargeConjectureissoverynatu ralgiventheproofofconditionalstatement(? ).Secondly, ,whodoesn tlikeachallenge;thelureofthecarrotisstro ng(butdon tdisregardthestick).Wecanapproachthispro blem:(1)purelymeasure-theoretically,(2)u singmeasure-theoreticergodicsystems,(3)u singdiscretetopologicaldynamicalsystems, (4)algebraicallythroughtheStone- CechcompactificationofZ+,and(5) , ,wemuststartwithSzemer edi s[19] Z+,let d(A)denotetheupperdensityofA: d(A)=limsupn!1|A\{1,2,..,n}| (Szemer edi stheorem).AnysubsetS Z+with d(S)> edi sproofhasbeencalledelementary,butitisany thingbuteasy,straightforward, , ,howdowemeshthisresultwith2-largesets?

7 Sinceevery2k-termarithmeticprogressionwi thcommondi erencedcontainsak-termarithmeticprogress ionwithcommondi erence2d,wehavelargesetswithpositivedens ity(thesetofevenpositiveintegers).Aresul tin[7]showsthat{10n:n2Z+}isnot2-large, , , +byalternatingredandblue, , edi sresult,BergelsonandLiebman[4] (butstillnotasgeneralasthefulltheorem), (BergelsonandLiebman).Letp(x):Z+!Z+beapo lynomialwithp(0)= {p(i):i2Z+} ,anysubsetofZ+ , edi sTheoremandergodicdynamicalsystemsisprov idedbyFurstenberg scorrespondenceprinciple[9], erentambientspaces;see, ,[5]. Z+andn2Z,weletS n={s n:s2S}.

8 Fortheremainderofthearticle,wereservethe symbolTfortheshiftoperatorthatactsonX,th efamilyofinfinitesequencesx=(xi)i2Z,byTx n=xn+ (Furstenberg sCorrespondencePrinciple).LetE Z+with d(E)> ,foranyk2Z+,thereexistsaprobabilitymeasu re-preservingdynamicalsystem(X,B, ,T)withasetA2 Bsuchthat (A)= d(E)and d k\i=0(E in)! k\i=0T inA!foranyn2Z+. +suchthat,foranyAwith (A)>0wehave A\T dA\T 2dA\ \T kdA > ,wehaveE\(E d)\(E 2d)\ \(E kd)6=;.Hence,bytakingainthisintersection ,wehave{a,a+d,a+2d,..,a+kd} ,FurstenbergprovidedanergodicproofofSzem er edi ; ,the2-LargeConjecturemaybesusceptibletot heuseofadi erentbreedofdynamicalsystem, (xn)n2 Zwithxi2{0,1.}

9 ,r 1} ,westateBirkho sMultipleRecurrenceTheoremduetoFurstenbe rgandWeiss[11](seealso[6]).Theorem6(Birk ho sMultipleRecurrenceTheorem).Letk,r2Z+.Un dertheproducttopology,foranyopensetU Xrthereexistsd2Z+sothatU\T dU\T 2d\ \T kdU6=;.Furstenberg sandWeiss resultallowedthemtogiveanewproofofvander Waerden sTheorem:Defineametricforx,y2 Xrbyd(x,y)= mini2Z+x(i)6=y(i) 1,wherex(i) (x,y) +.Theorem6helpsprovethatthereexistsy2{Tm x}m2Z+suchthatallofd(y,Tdy),d(y,T2dy),.. ,d(y,Tkdy) ,y,Tdy,T2dy,.., ,xa+d,xa+2d,..,xa+kdallofthesamevalue/co lor,meaningthata,a+d,..,a+ (y,Tdy),d(y,T2dy).

10 ,d(y,Tkdy)arelessthanany >0;however,thisisnotneededtoprovevanderW aerden , ,wehaveaguaranteethatoverthespaceX2there existsy2{Tmx}m2Z+suchthatallofd(y,Tdy),d (y,T2dy),..,d(y,Tkdy) (y,Tdy),d(y,T2dy),..,d(y,Tkdy)canbearbit rarilysmall,wecanonlyguaranteetheyareles sthan1(withourgivenmetric) ,wecouldconvertanr-coloringtoabinaryequi valent2-coloringifwediscoveredaresulttha talongenough(r 1)-coloredD-apadmitsamonochromatick-term D-ap(wehavenosuchresult,butthisideawillp rovefruitfulinSection6). , , ,beforegettingtotheStone- Cechcompactification,we llhaveadiagramtoaidinvisualizinghowthedi ,wegiveimplicationsbetweenthetypesofrecu rrencewehaveconsideredthusfar, ;assuch, +.