Journal of Applied Analysis Vol. 9, No. 2 (2003), pp. 149–162 CRITICAL CARDINALITIES AND A

CRITICALCARDINALITIESANDADDITIVITYPROPERTIESOF

COMBINATORIALNOTIONSOFSMALLNESS

S.SHELAHandB.TSABAN

ReceivedAugust6,2002and,inrevisedform,May19,2003

Abstract.Motivatedbytheminimaltowerproblem,anearlierworkstudieddiagonalizationsofcoverswherethecoversarerelatedtolin-earquasiorders(τ-covers).Wedealwithtwotypesofcombinatorialquestionswhicharisefromthisstudy.

1.Twonewcardinalsintroducedinthetopologicalstudyareex-pressedintermsofwellknowncardinalscharacteristicsofthecon-tinuum.

2.Westudytheadditivitynumbersofthecombinatorialnotionscor-respondingtothetopologicaldiagonalizationnotions.

ThisgivesnewinsightsonthestructureoftheeventualdominanceorderingontheBairespace,thealmostinclusionorderingontheRoth-bergerspace,andtheinteractionsbetweenthem.

2000MathematicsSubjectClassification.03E17,06A07,03E35,03E10.

Keywordsandphrases.τ-cover,tower,splittingnumber,additivitynumber.

TheresearchofthefirstauthorispartiallysupportedbyTheIsraelScienceFoundationfoundedbytheIsraelAcademyofSciencesandHumanities.Publication768.

Thispaperconstitutesapartofthesecondauthor’sdoctoraldissertationatBar-IlanUniversity.

ISSN1425-6908cHeldermannVerlag.󰀁

150S.SHELAHandB.TSABAN

1.Introductionandoverview

Letωdenotethesetofnaturalnumbers.Weworkwithtwospaceswhichcarryaninterestingcombinatorialstructure:TheBairespaceωωwitheventualdominance≤∗(f≤∗giff(n)≤g(n)forallbutfinitelymanyn),andtheRothbergerspace[ω]ω={A⊆ω:Aisinfinite}with⊆∗(A⊆∗BifA\\Bisfinite).WewriteA⊂∗BifA⊆∗BandB⊆∗A.

AsubsetXofωωisunboundedifitisunboundedwithrespectto≤∗.Xisdominatingifitiscofinalinωωwithrespectto≤∗.bistheminimalsizeofanunboundedsubsetofωω,anddistheminimalsizeofadominatingsubsetofωω.

AninfinitesetA⊆ωisapseudo-intersectionofafamilyF⊆[ω]ωifforeachB∈F,A⊆∗B.AfamilyF⊆[ω]ωisatowerifitislinearlyquasiorderedby⊆∗,andithasnopseudo-intersection.tistheminimalsizeofatower.AfamilyF⊆[ω]ωiscenterediftheintersectionofeach(nonempty)finitesubfamilyofFisinfinite.pistheminimalsizeofacenteredfamilywhichhasnopseudo-intersection.AfamilyF⊆[ω]ωissplittingifforeachinfiniteA⊆ωthereexistsS∈FwhichsplitsA,thatis,suchthatthesetsA∩SandA\\Sareinfinite.sistheminimalsizeofasplittingfamily.

Letc=2ℵ0.Thefollowingrelations,whereanarrowmeans≤,arewell-known[3]:

b󰀋󰀌

ℵ1→p→td→c

󰀌󰀋

sNopairofcardinalsinthisdiagramisprovablyequal,exceptperhapspandt.TheMinimalTowerproblem,whichaskswhetheritisprovablethatp=t,isoneofthemostimportantproblemsininfinitecombinatorics,anditgoesbacktoRothberger(see,e.g.,[12]).

Newcardinals.In[15],topologicalnotionsrelatedtopandtwerecom-pared.In[17]thetopologicalnotionrelatedtot(calledτ-covers)wasstud-iedinawidercontext.Thisstudyledbacktoseveralnewcombinatorialquestions,oneofwhichrelatedtotheminimaltowerproblem.

Definition1.ForafamilyF⊆[ω]ωandaninfiniteA⊆ω,defineF󰀖A={B∩A:B∈F}.IfallsetsinF󰀖Aareinfinite,wesaythatF󰀖AisalargerestrictionofF.LetκωτbetheminimalcardinalityofacenteredfamilyF⊆[ω]ωsuchthatthereexistsnoinfiniteA⊆ωsuchthattherestrictionF󰀖Aislargeandlinearlyquasiorderedby⊆∗.

COMBINATORIALNOTIONSOFSMALLNESSANDADDITIVITY151

Itisnotdifficulttoseethatp=min{κωτ,t}[17].InSection2weshowthatinfact,p=κωτ.Thisexistenceofacenteredfamilywithnolargelinearlyquasiorderedrestrictionshowsthatpiscombinatorially“larger”thanassertedinitsoriginaldefinition,andsuggestsanadditionalevidencetothedifficultyofseparatingpfromthecombinatorially“larger”cardinalt:NowtheconsistencyofκωτDefinition2.Forfunctionsf,g∈ωω,andabinaryrelationRonω,defineasubset[fRg]ofωby:

[fRg]={n:f(n)Rg(n)}.

Next,Forfunctionsf,g,h∈[hRgSf]⊆ωby:

ωω,

andbinaryrelationsR,Sonω,define

[fRgSh]=[fRg]∩[gSh]={n:f(n)Rg(n)andg(n)Sh(n)}.ForasubsetXofωωandg∈ωω,wesaythatgavoidsmiddlesinXwithrespectto󰀜R,S󰀝if:

1.foreachf∈X,theset[fRg]isinfinite;

2.forallf,h∈Xatleastoneofthesets[fRgSh]and[hRgSf]isfinite.

Xsatisfiestheexcludedmiddlepropertywithrespectto󰀜R,S󰀝ifthereexistsg∈ωωwhichavoidsmiddlesinXwithrespectto󰀜R,S󰀝.xR,SistheminimalsizeofasubsetXofωωwhichdoesnotsatisfytheexcludedmiddlepropertywithrespectto󰀜R,S󰀝.

Thecardinalx=x<,≤wasdefinedin[17].InSection3weexpressallofthefourcardinalsx≤,≤,x<,≤,x≤,<,andx<,Additivityofcombinatorialnotionsofsmallness.ForafinitesubsetFofωω,definemax(F)∈ωωbymax(F)(n)=max{f(n):f∈F}foreachn.AsubsetYofωωisfinitely-dominatingifthecollection

maxfin(Y):={max(F):FisafinitesubsetofY}

isdominating.

Wewillusethefollowingnotations:

B:thecollectionofallboundedsubsetsofωω;

X:thecollectionofallsubsetsofωωwhichsatisfytheexcludedmiddlepropertywithrespectto󰀜<,≤󰀝;

Dfin:thecollectionofallsubsetsofωωwhicharenotfinitelydominating;

and

D:thecollectionofallsubsetsofωωwhicharenotdominating.

152S.SHELAHandB.TSABAN

ThusB⊆X⊆Dfin⊆D.TheclassesB,X,Dfin,andDareusedtocharacterizecertaintopologicaldiagonalizationproperties[13,16,17].

Following[1],wedefinetheadditivitynumberforclassesI⊆J⊆P(ωω)with∪I∈Jby

add(I,J)=min{|F|:F⊆Iand∪F∈J},

andwriteadd(J)=add(J,J).IfIcontainsallsingletons,thenadd(I,J)≤non(J),wherenon(J)=min{|J|:J⊆ωωandJ∈J}(thusnon(B)=b,non(D)=non(Dfin)=d,andnon(X)=x.)

ForI,J∈{B,X,Dfin,D},thecardinalsadd(I,J)boundfrombelowtheadditivitynumbersofthecorrespondingtopologicaldiagonalizations.InSection4weexpressadd(I,J)foralmostallI,J∈{B,X,Dfin,D}intermsofwellknowncardinalcharacteristicsofthecontinuum.Intwocasesforwhichthisisnotdone,wegiveconsistencyresults.

2.Thecardinalκωτ

Forourpurposes,afilteronabooleansubalgebraBofP(ω)isafamilyU⊆BwhichisclosedundertakingsupersetsinBandfiniteintersections,anddoesnotcontainfinitesetsaselements.Theorem3.p=κωτ.

Proof.LetF⊆[ω]ωbeacenteredfamilyofsizepwhichhasnopseudo-intersection.LetBbethebooleansubalgebraofP(ω)generatedbyF.Then|B|=p.LetU⊆BbeafilterofBcontainingF.AsUdoesnotcontainfinitesetsaselements,Uiscentered.Moreover,|U|=p,andithasnopseudo-intersection.

Towardsacontradiction,assumethatp<κωτ.ThenthereexistsaninfiniteA⊆ωsuchthattherestrictionU󰀖Aislarge,andislinearlyquasiorderedby⊆∗.FixanyelementD0∩A∈U󰀖A.AsU󰀖Adoesnothaveapseudo-intersection,thereexist:

1.AnelementD1∩A∈U󰀖AsuchthatD1∩A⊂∗D0∩A;and2.AnelementD2∩A∈U󰀖AsuchthatD2∩A⊂∗D1∩A.

Thenthesets(D2∪(D0\\D1))∩AandD1∩A(whichareelementsofU󰀖A)containtheinfinitesets(D0∩A)\\(D1∩A)and(D1∩A)\\(D2∩A),respectively,andthusarenot⊆∗-comparable,acontradiction.Acloselyrelatedproblemfrom[17]remainsopen.

COMBINATORIALNOTIONSOFSMALLNESSANDADDITIVITY153

Definition4.AfamilyY⊆[ω]ωislinearlyrefinableifforeachy∈Y

ˆ={ythereexistsaninfinitesubsetyˆ⊆ysuchthatthefamilyYˆ:y∈Y}

∗∗islinearly⊆-quasiordered.pistheminimalsizeofacenteredfamilyin

[ω]ωwhichisnotlinearlyrefineable.

Againitiseasytoseethatp=min{p∗,t}.Thus,asolutionofthefollowingproblemmayshedmorelightontheMinimalTowerproblem.Problem5.Doesp=p∗?

3.Theexcludedmiddleproperty

Lemma6.b≤x≤,≤≤x≤,<≤x<,≤≤x<,<≤d.

Proof.Theinequalitiesx≤,≤≤x≤,AssumethatYisaboundedsubsetofωω.Letg∈ωωboundY.ThengavoidsmiddlesinYwithrespectto󰀜≤,≤󰀝.Thisshowsthatb≤x≤,≤.

Next,considerasubsetYofωωwhichsatisfiestheexcludedmiddleprop-ertywithrespectto󰀜<,<󰀝,andletgwitnessthat.ThengwitnessesthatYisnotdominating.Thusx<,<≤d.

Itremainstoshowthatx≤,<≤x<,≤.AssumethatY⊆ωωsatisfiestheexcludedmiddlepropertywithrespectto󰀜≤,<󰀝,andletg∈ωωavoidmiddlesinYwithrespectto󰀜≤,<󰀝.Defineg˜∈ωωsuchthatg˜(n)=g(n)+1foreachn.Foreachf,h∈Ywehavethat[f≤g]=[fTheorem7.x≤,≤=x≤,<=b.

Proof.ByLemma6,itisenoughtoshowthatx≤,<≤b.Let󰀜bα:α1ωbeanunboundedsubsetofωω.Foreachα󰀂󰀂0bα(2n)=bα(n)b1=0α(2n);b0b1α(2n+1)=0α(2n+1)=bα(n)

1foreachn∈ω,andsetY={b0α,bα:αthatYdoesnotsatisfytheexcludedmiddlepropertywithrespectto󰀜≤,<󰀝.Foreachg∈ωω,letα101

[b0α≤g⊇{2n+1:0≤g(2n+1)154S.SHELAHandB.TSABAN

0isaninfiniteset.Similarly,[b1α≤ginfinite.Thatis,gdoesnotavoidmiddlesinYwithrespectto󰀜≤,<󰀝.

Lemma8.s≤x<,≤.

Proof.AssumethatY⊆ωωissuchthat|Y|Consideranyset[f[f⊆{n∈A:f(n)⊆{n∈A:h(n)≤f(n)}=A\\[fisfinite.Otherwise,A\\[fisfinite.ThusYsatisfiestheexcludedmiddlepropertywithrespectto󰀜<,≤󰀝.

Theorem9.x<,≤=x<,<=max{s,b}.

Proof.ByLemmas6and8,wehavethatmax{s,b}≤x<,≤≤x<,<.Wewillprovethatx<,<≤max{s,b}.TheargumentisanextensionoftheproofofTheorem7.

Letb∗betheminimalsizeofasubsetBofωωsuchthatBisunboundedoneachinfinitesubsetofω.Accordingto[3],b=b∗.ThusthereexistsasubsetB=󰀜bα:α1ωbeasplittingfamily.Foreachα󰀂󰀂bβ(n)n∈Sα0n∈Sα1

b0(n)=;b(n)=α,βα,β

0n∈Sαbβ(n)n∈SαandsetY={biα,β:i<2,αWewillshowthatYdoesnotsatisfytheexcludedmiddlepropertywithrespectto󰀜<,<󰀝.Assumethatg∈ωωavoidsmiddlesinYwithrespectto󰀜<,<󰀝.ThenthesetA=[0COMBINATORIALNOTIONSOFSMALLNESSANDADDITIVITY155

bγ󰀖A∩Sα≤∗g󰀖A∩Sα,andβ>γsuchthatbβ󰀖A\\Sα≤∗g󰀖A\\Sα.Then

101

[b0α,β={n∈A\\Sα:0={n∈A\\Sα:g(n)isaninfiniteset.Similarly,theset

010

[b1α,β={n∈A∩Sα:0={n∈A∩Sα:g(n)isalsoinfinite,becausebγ≤∗bβ;acontradiction.

Remark10.Thecardinalmax{s,b}isalsoequaltothefinitelysplittingnumberfsstudiedin[8].

Severalvariationsoftheexcludedmiddlepropertyarestudiedintheappendixtotheonlineversionofthispaper[14].

4.Additivityofcombinatorialproperties

Theadditivitynumberadd(I,J)ismonotonedecreasinginthefirstcoor-dinateandincreasinginthesecond.Ourtaskinthissectionistodetermine,whenpossible,thecardinalsinthefollowingdiagramintermsoftheusualcardinalcharacteristicsb,d,etc.(Inthisdiagram,anarrowmeans≤.)

add(D,D)→

add(Dfin,D)→add(X,D)→add(B,D)

↑↑↑

add(Dfin,Dfin)→add(X,Dfin)→add(B,Dfin)

↑↑add(X,X)→add(B,X)

↑

add(B,B)

4.1.ResultsinZFC.

Theorem11.Thefollowingequalitieshold:1.add(B,Dfin)=add(B,D)=d;

2.add(Dfin,Dfin)=add(X,X)=add(X,Dfin)=2;and3.add(D,D)=add(B,B)=add(B,X)=b.

156S.SHELAHandB.TSABAN

Proof.(1)Asnon(D)=d,itisenoughtoshowthatadd(B,Dfin)≥d.󰀁

Assumethat|I|˜benotdominating;lethbeawitness󰀁forthat.ForeachfiniteF⊆Y,letI

∗afinitesubsetofIsuchthatF⊆i∈I˜Yi.Thenmax(F)≤max({gi:i∈

˜})≥∗h.Thusmax(F)≥∗h,soY∈Dfin.I

(2)Itisenoughtoshowthatadd(X,Dfin)=2.Thus,let

Y0={f∈ωω:(∀n)f(2n)=0andf(2n+1)≥1}Y1={f∈ωω:(∀n)f(2n)≥1andf(2n+1)=0}.

Thentheconstantfunctiong≡1witnessesthatY0,Y1∈X,butY0∪Y1is2-dominating,andinparticularfinitelydominating.

(3)Itisfolklorethatadd(D,D)=add(B,B)=b–see,e.g.,[2,fullversion]foraproof.Itremainstoshowthatadd(B,X)≤b.LetBbeasubsetofωωwhichisunboundedoneachinfinitesubsetofω,andsuchthat|B|=b.Foreachf∈BletYf󰀁={g∈ωω:g≤∗f}.(ThuseachYfisbounded.)WeclaimthatY=f∈BYf∈X.Tothisend,consideranyfunctiong∈ωωwhichclaimstowitnessthatY∈X.Inparticular,[04.2.Consistencyresults.

Theonlycaseswhichwehavenotsolvedyetareadd(Dfin,D)andadd(X,D).In[2,fullversion]itwasprovedthatb≤add(Dfin,D).InTheorem2.2of[10]itis(implicitly)provedthatg≤add(Dfin,D).Thus

max{b,g}≤add(Dfin,D)≤add(X,D)≤d.

Moreover,foranyI⊆J,cf(add(I,J))≥add(J),andtherefore

cf(add(Dfin,D)),cf(add(X,D))≥add(D,D)=b.

Thenotionofultrafilterwillbeusedtoobtainupperboundsonadd(Dfin,D)andadd(X,D).AfamilyU⊆[ω]ωisanonprincipalultrafilterifitisclosedundertakingsupersetsandfiniteintersections,andcannotbeextended,thatis,foreachinfiniteA⊆ω,eitherA∈Uorω\\A∈U.Consequently,alinearquasiorder≤Ucanbedefinedonωωby

f≤Ug

if

[f≤g]∈U.

COMBINATORIALNOTIONSOFSMALLNESSANDADDITIVITY157

Thecofinalityofthereducedproductωω/UistheminimalsizeofasubsetCofωωwhichiscofinalinωωwithrespectto≤U.

Theorem12.Foreachcardinalnumberκ,thefollowingareequivalent:1.κ2.foreachκ-sequence󰀜(gα,Uα):α<κ󰀝witheachUαanultrafilteronωandeachgα∈ωωthereexistsg∈ωωsuchthatforeachα<κ,[gα≤g]∈Uα.Proof.1⇒2:Foreachα<κ󰀁letYα={f∈ωω:[f󰀁

2⇒1:AssumethatY=α<κYαwhereeachYα∈Dfin.Foreachα,letUαbeanultrafiltersuchthatYα/Uαisbounded,saybygα∈ωω[13].By(2)letg∈ωωbesuchthatforeachα<κ,[gα≤g]∈Uα.ThengwitnessesthatYisnotdominating:Foreachf∈Y,letαbesuchthatf∈Yα.Then[f≤gα]∈Uα,thus[fCorollary13.AssumethatUisanonprincipalultrafilteronω.Thenadd(Dfin,D)≤cof(ωω/U).

Proof.AssumethatκProof.Canjar[7]provedthatthereexistsanonprincipalultrafilterUwithcof(ωω/U)=cf(d).NowuseCorollary13.

Lemma15.g∈ωωavoidsmiddlesinYif,andonlyif,foreachf∈Y[fTheorem16.Foranycardinalκ,thefollowingareequivalent:1.κ158S.SHELAHandB.TSABAN

2.foreachκ-sequence󰀜(gα,Fα):α<κ󰀝,suchthateachgα∈ωω,andforeachαtherestrictionFα󰀖[0Proof.2⇒1:AssumethatY=α<κYαwhereeachYα∈X.Foreachαletgα∈ωωbeafunctionavoidingmiddlesinYα,andsetFα={[f1⇒2:ReplacingeachFαwithFα󰀖[0󰀂

n∈A˜(n)=gα(n)−1h

max{gα(n),h(n)}otherwise.˜AsFαislinearlyquasiordered󰀁by⊆∗,wehavebyLemma15thatgαavoidsmiddlesinYα.By(1),Y=α<κYαisnotdominating;leth∈ωωbeawitnessforthat.

˜∈YαbethefunctiondefinedinForeachα<κandA∈Fα,leth

˜∈Y,therefore[h˜˜,[h˜AnonprincipalultrafilterUisasimplePκpointifitisgeneratedbyaκ-sequence󰀜Aα:α<κ󰀝⊆[ω]ωwhichisdecreasingwithrespectto⊆∗.Uisapseudo-PκpointifeveryfamilyF⊆Uwith|F|<κhasapseudo-intersection.ClearlyeverysimplePκpointisapseudo-Pκpoint.Corollary17.IfUisasimplePκpoint,thenadd(X,D)≤cof(ωω/U).Proof.AssumethatλFα={[h(4.1)

COMBINATORIALNOTIONSOFSMALLNESSANDADDITIVITY159

Wemayassumethatforeachα<λ,[0Intheremainingpartofthepaperwewillconsidertheremainingstandardcardinalcharacteristicsofthecontinuum(see[3]).Letudenotetheminimalsizeofanultrafilterbase.

Theorem18.Itisconsistent(relativetoZFC)thatthefollowingholds:

u=add(Dfin,D)=add(X,D)=ℵ1<ℵ2=s=c.

Thus,itisnotprovablethats≤add(X,D).

Proof.In[5]amodelofsettheoryisconstructedwherec=ℵ2andthereexistasimplePℵ1pointandasimplePℵ2point.ThesimplePℵ1pointisgeneratedbyℵ1manysets,thusu=ℵ1.Asb≤u,b=ℵ1aswell.

NyikosprovedthatifthereexistsapseudoPκpointUandκ>b,thencof(ωω/U)=b(see[4]).ThusbyCorollary17,add(X,D)≤b=ℵ1inthismodel.In[4]itisprovedthatifthereexistsapseudoPκpointU,thens≥κ.Therefores≥ℵ2inthismodel.

Depth+([ω]ω)isdefinedastheminimalcardinalκsuchthatthereexistsno⊂∗-decreasingκ-sequencein[ω]ω.(Thus,e.g.,t1.IfDepth+([ω]ω)Proof.Toprove(1)itisenoughtoshowthatforeachκsatisfyingDepth+([ω]ω)≤κAssumethatDepth+([ω]ω)≤κ160S.SHELAHandB.TSABAN

UsethefactthatDepth+([ω]ω)≤κ(respectively,Depth+([ω]ω)=d)to

˜αofFαsuchthat|F˜α|<κchooseforeachα<κacofinalsubfamilyF

˜α|Wemayassumethateachgαisincreasing.ForeachαandeachA∈Fα,󰀢∈ωωbetheincreasingenumerationofA.Thecollection{gα◦A󰀢:letA

α<κ,A∈Fα}haslessthandmanyelementsandthereforecannotbedominating.Leth∈ωωbeawitnessforthat.Fixα<κ.ForallA∈Fα,thereexistinfinitelymanynsuchthat

󰀢(n))=gα◦A󰀢(n)thatis,A∩[gαTheorem20.AssumethatVisamodelofCHandℵ1<κ=κℵ0.Let

CκbetheforcingnotionwhichadjoinsκmanyCohenrealstoV.ThenintheCohenmodelVCκ,thefollowingholds:

add(Dfin,D)=s=a=non(M)=ℵ1In[6,11]itisprovedthatthereexistsanonprincipalultrafilterUinVCκsuchthatcof(ωω/U)=ℵ1.ByCorollary13,wehavethatadd(Dfin,D)=ℵ1inVCκ.

Inparticular,thecardinalsadd(Dfin,D)andadd(X,D)arenotprovablyequal.

Corollary21.Itisnotprovablethatadd(X,D)≤cf(d).

Proof.UseTheorem20withκ=ℵℵ1.InVCκ,d=c=ℵℵ1,thereforecf(d)=ℵ1Remark22.Intheremainingcanonicalmodelsofsettheorywhichareusedtodistinguishbetweenthevariouscardinalcharacteristicsofthecontin-uum(see[3]),max{b,g}=dholds,andthereforeadd(Dfin,D)=add(X,D)=

COMBINATORIALNOTIONSOFSMALLNESSANDADDITIVITY161

dtoo.Thesemodelsshowthatnoneofthefollowingisprovable:

min{cov(N),r}≤add(X,D)(Randomrealsmodel),add(Dfin,D)≤max{cov(N),s}(Hechlerrealsmodel),add(Dfin,D)≤max{non(N),cov(N)}(Laverrealsmodel),andadd(Dfin,D)≤max{u,a,non(N),non(M)}(Millerrealsmodel).Collectingalloftheconsistencyresults,wegetthattheonlypossibleadditionallowerboundsonadd(X,D)arecov(M)ande(observethate≤cov(M)[3].)

Problem23.Iscov(M)≤add(X,D)?Andifnot,ise≤add(X,D)?Noadditionalcardinalcharacteristiccanserveasanupperboundonadd(Dfin,D).

Anotherquestionofinterestiswhetheradd(Dfin,D)oradd(X,D)appearinthelatticegeneratedbythecardinalcharacteristicswiththeoperationsofmaximumandminimum.Inparticular,wehavethefollowing.Problem24.Isitprovablethatadd(Dfin,D)=max{b,g}?

WehaveanindicationthattheanswertoProblem24isnegative,butthisisadelicatematterwhichwillbetreatedinafuturework.

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SaharonShelah

InstituteofMathematics

HebrewUniversityofJerusalemGivatRam,91904JerusalemIsrael

andMathematicsDepartmentRutgersUniversity

NewBrunswick,NJ08903USA

shelah@math.huji.ac.il

BoazTsaban

DepartmentofMathematics

andComputerScienceBar-IlanUniversityRamat-Gan52900,Israel

tsaban@macs.biu.ac.il