coloringc0X2de nedbyc0xcxifx2XnY0dxifx2Y0satis esFriendc0Friendc2Y0FriendcFriendGcontradictingthede nitionofFriendG Remark22Whiletheminimizationof2friendlinessissucien ID: 205344
Download Pdf The PPT/PDF document "UsingnowastandardBorelspaceXasourvertexs..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
UsingnowastandardBorelspaceXasourvertexset,wesayagraphGonXisBorelifitisBorelasa(symmetric,irre exive)subsetofX2.Itwillbemoreconvenienttousethelanguageofcoloringsratherthanthatofpartitions.Towardsthatend,givenn2N+and 2[0;1]wesaythatc:X!nisan(n; )-coloringofalocallynitegraphGonXifforallx2Xwehavejc�1(c(x))\Gxj jGxj,whereGxdenotesthesetofneighborsofx.Soan(n;0)-coloringiswhatisnormallycalledapropern-coloring,i.e.,notwoadjacentverticesreceivethesamecolor.Furthermore,cisa(2;1=2)-coloringi c�1(0)tc�1(1)formsanunfriendlypartition.IfmoreoverisaBorelprobabilitymeasureonX,wedene(n; ;)-coloringstobethosefunctionssatisfyingtheconditionjc�1(c(x))\Gxj jGxjona-conullsetofx2X.OurfocusinthispaperisontheexistenceofBorel(n; ;)-coloringsforvariousclassesofBorelgraphsonX.Inparticular,supposethat�isagroupwithnite,symmetricgeneratingsetS(whichwealwaysassumedoesnotcontaintheidentity).Associatedwithanyfree,-preservingBorelactionof�on(X;)isagraphrelatingdisctinctpointsofXifandonlyifanelementofSsendsonetotheother.Wethenhave(seesection3foradenitionofweakequivalence)Theorem1.1.Supposethat(X;)isastandardprobabilityspace,n2N+,and�isagroupwithnite,symmetricgeneratingsetS.Thenanyfree,-preservingactionBorelactionof�on(X;)isweaklyequivalenttoonewhoseassociatedgraphadmitsaBorel(n;1=n;)-coloring.Recallthatthe(right)Cayleygraph,Cay(�;S)ofagroup�withdes-ignatedgeneratingsetShasvertexset�andedges( ; s)for 2�ands2S.Wemayviewthespaceof(n; )-coloringsofCay(�;S)asasubsetofn�whichisclosedintheproducttopology,soacompactPolishspaceinitsownright.Then�actsby(left)translationsonthespaceof(n; )-coloringsby( c)()=c( �1).Theorem1.2.Supposethat�isagroupwithnite,symmetricgeneratingsetS.Thenthereisatranslation-invariantBorelprobabilitymeasureonthespaceof(n;1=n)-coloringsoftheCayleygraphCay(�;S).Suchameasuremaybeviewedasa(translation-invariant)random(n;1=n)-coloringofCay(�;S).Inparticular,inthecasen=2weobtainarandomunfriendlypartitionoftheCayleygraph.2 coloringc0:X!2denedbyc0(x)=(c(x)ifx2XnY0d(x)ifx2Y0satisesFriend(c0)Friend(c)�2(Y0)Friend(c)=Friend(G),contra-dictingthedenitionofFriend(G). Remark2.2.Whiletheminimizationof2-friendlinessissucientforacol-oringtoinduceanunfriendlypartition,itisfarfromnecessary.Forinstance,considerforxedirrational 2(0;1)thegraphG on[0;1)wherexG yi x�y= mod1.ThenFriend2(G )=0,butitisnothardtoshowthatFriend2(c)�0foreachBorelc:[0;1)!2.Nevertheless,the2-regularityofG makesitstraightforwardtondaBorelunfriendlypartitionforG :in-deed,by[5,Proposition4.2]thereisamaximalG independentsetwhichisBorel,soitanditscomplementformanunfriendlypartition.Intheinterestoffulldisclosure,weactuallydon'tknowwhethereverylocallyniteBorelgraphadmitsaBorelunfriendlypartition.Question2.3.SupposethatGisalocallynite,-preservinggraphon(X;)andn2N+.DoesGadmitaBorel(n;1=n)-coloring,oratleastaBorel(n;1=n;)-coloring?Theresultsofthenextsectionruleoutcertainpossiblecounterexamplesarisingfromcombinatorialinformationinvariantunderweakequivalenceofgroupactions.3GroupactionsWenextnarrowourfocustographsarisingfromgraphsarisingfromcount-ablegroupsactingbyfree-preservingautomorphismson(X;).Givenacountablegroup�withsymmetricgeneratingsetSandsuchanactionaof�on(X;),wedenethegraphG(S;a)onthevertexsetXwithedge(x;y)i x6=yand9s2S(y=sx).Byfreenessoftheaction(andassumingthatSdoesnotcontaintheidentityelementofthegroup),eachvertexhasdegreejSj,soifSisanitegeneratingsetthegraphislocallynite(andinfacthasboundeddegree).CombinatorialparametersassociatedwithG(S;a)re ectvariousdynamicalpropertiesoftheactiona;formoresee[2].4 inwesee(G(S;b)d�1(i))=Xs2S(d�1(i)\sbd�1(i))Xs2S((Bi\sbBi)+2")Xs2S((Ai\saAi)+3")=(G(S;a)Ai)+3jSj":Consequently,Friendn(d)=Xi(G(S;b)d�1(i))Xi(G(S;a)Ai)+3jSj"=Friendn(c)+3njSj"Friendn(G(S;a))+(3njSj+1)":As"maybechosentobearbitrarilysmall,weseethatFriendn(G(S;a))Friendn(G(S;b))asdesired. Corollary3.2.Supposethat�isagroupwithnite,symmetricgenerat-ingsetS,anda;b2FR(�;X;)withab.ThenFriendn(G(S;a))=Friendn(G(S;b)).Next,werecordaversionof[3,Theorem5.2]allowingustorealizetheinmuminthedenitionoffriendlinesswithinanyweakequivalenceclass.Proposition3.3.Supposethat�isagroupwithnite,symmetricgenerat-ingsetS.Foranya2FR(�;X;)thereisanactionb2FR(�;X;)withbaanda-measurablefunctionc:X!nsuchthatFriendn(G(S;b))=Friendn(c).Proof.Weusethenotationof[3].LetUbeanonprincipalultralteronN.WeobtaintheactionbasanappropriatelychosenfactoroftheultrapoweractionaUon(XU;U).Putf=Friendn(G(S;a))andxfork2NBorelfunctionsck:X!nsatisfyingFriendn(ck)j+1=k.Putforeachin,Ci=[c�1k(i)]U.ThenfCi:ingformaU-a.e.partitionofXU.Dening6 Inparticular,atranslation-invariantrandom(2;1=2)-coloringmaybeviewedasarandomunfriendlypartitionoftheCayleygraph,wherethetranslationinvariancemeansthatthelikelihoodofchoosingapartitionisindependentoftheselectionofavertexoftheCayleygraphastheidentity.Corollary4.1.Supposethat�isagroupwithnite,symmetricgeneratingsetSandn2N+.Thenthereisatranslation-invariantrandom(n;1=n)-coloringofCay(�;S).Proof.Fixanonatomicstandardprobabilityspace(X;).ByTheorem3.4thereissomeb2FR(�;X;)suchthatG(S;b)admitsaBorel(n;1=n;)-coloringc:X!n(infactbmaybechosenfromanyweakequivalenceclass).Dene:X!Col(�;S;n;1=n)by((x))( )=c( �1x).Thenisatranslation-invariantrandom(n;1=n)-coloring,whereasusual(A)=(�1(A)). WeclosewithaquestionwhichisessentiallyaprobabilisticversionofQuestion3.5.Question4.2.Cansuchatranslation-invariantrandom(n;1=n)-coloringbefoundasafactorofIID?Acknowledgments.TheauthorwouldliketothankAlekosKechris,RussLyons,AndrewMarks,BenMiller,JustinMoore,andRobinTucker-Drobforusefulconversations.SpecialthanksgotoSimonThomaswhorelayedthequestionaboutdenableunfriendlypartitionsduringMAMLS2012atRutgers.References[1]R.Aharoni,E.C.Milner,andK.Prikry.Unfriendlypartitionsofagraph.J.Combin.TheorySer.B,50(1):1{10,1990.[2]C.T.ConleyandA.S.Kechris.Measurablechromaticandindependencenumbersforergodicgraphsandgroupactions.GroupsGeom.Dyn.,7(1):127{180,2013.[3]C.T.Conley,A.S.Kechris,andR.D.Tucker-Drob.Ultraproductsofmea-surepreservingactionsandgraphcombinatorics.Erg.TheoryandDy-nam.Systems.Toappear.8