Contents lists available atScienceDirect
Advances in Mathematics
www.elsevier.com/locate/aim
Homogeneous families on trees and subsymmetric basic sequences
✩,✩✩C. Brecha, J. Lopez-Abadb,c,∗,S. Todorcevicd,e
a DepartamentodeMatemática,InstitutodeMatemáticaeEstatística, UniversidadedeSãoPaulo,CaixaPostal66281,05314-970,SãoPaulo,Brazil
b InstitutdeMathématiquesdeJussieu–PRG,UniversitéParis7,case7012, 75205 ParisCedex 13,France
cDepartamentodeMatemáticasFundamentales,UNED,PaseoSendadelRey9, E-28040Madrid,Spain
d InstitutdeMathématiquesdeJussieu,UMR7586,2placeJussieu –Case247, 75222ParisCedex05,France
eDepartmentofMathematics,UniversityofToronto,Toronto,M5S2E4,Canada
a r t i c l e i n f o a bs t r a c t
Article history:
Received24July2016
Receivedinrevisedform22March 2018
Accepted19June2018 Availableonlinexxxx
CommunicatedbySlawomirSolecki
MSC:
03E05 05D10 46B15 46B06 46B26
WestudydensityrequirementsonagivenBanachspacethat guarantee theexistenceofsubsymmetricbasicsequencesby extending Tsirelson’swell-known spaceto largerindex sets.
WeprovethatforeverycardinalκsmallerthanthefirstMahlo cardinalthereisareflexiveBanachspaceofdensityκwithout subsymmetric basicsequences. Asfor Tsirelson’sspace, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate thecomplexityon any giveninfinite set. This isusedtodescribedetailedlytheasymptoticstructureofthe spaces.Thecollectionsoffamiliesareofindependentinterest and their existence is proved inductively. The fundamental
✩ ThefirstauthorwaspartiallysupportedbyFAPESPgrants(2012/24463-7and2015/26654-2)andby CNPqgrants(307942/2012-0and454112/2015-7).ThesecondauthorwaspartiallysupportedbyFAPESP grant (2013/24827-1), by the Ministerio de Economía y Competitividad under grant MTM2012-31286 (Spain). The firstandthe thirdauthorwere partially supportedbyUSP-COFECUB grant (31466UC).
ThethirdauthorwasalsopartiallysupportedbygrantsfromNSERC(455916)andCNRS(UMR7586).
✩✩ ThisresearchwaspartiallydonewhilsttheauthorswerevisitingfellowsattheIsaacNewtonInstitute forMathematicalSciencesintheprogramme“Mathematical,FoundationalandComputationalAspectsof theHigherInfinite”(HIF).
* Correspondingauthor.
E-mailaddresses:brech@ime.usp.br(C. Brech),abad@mat.uned.es(J. Lopez-Abad), stevo@math.toronto.edu(S. Todorcevic).
https://doi.org/10.1016/j.aim.2018.06.008 0001-8708/©2018ElsevierInc. Allrightsreserved.
Keywords:
Familiesoffinitesets Spreadingmodels
NonseparableBanachspaces
steppingup argumentis the analysisof such collections of familiesontrees.
©2018ElsevierInc.Allrightsreserved.
1. Introduction
Recall that a set of indiscernibles for a given structure M is a subset X with a total ordering < such that for every positive integer n every two increasing n-tuples x1 < x2 <·· ·< xn and y1< y2<·· ·< yn of elementsof X havethesameproperties in M. A simple way of finding an extended structure on κ without an infinite set of indiscernibles is as follows. Suppose that F is a family of finite subsets of κ that is compact, as anaturalsubset of theproduct space 2κ, and large,that is, everyinfinite subsetofκhasarbitrarilylargesubsetsinF. LetMF bethestructure(κ,(F ∩[κ]n)n) thathasκasuniverseandthathasinfinitelymanyn-aryrelationsF ∩[κ]n ⊆[κ]n.Itis easilyseenthatMF doesnothaveinfiniteindiscerniblesets.
Whileinsettheoryandmodeltheoryindiscernibilityisawell-studiedandunambigu- ousnotion,inthecontext of theBanachspace theoryit hasseveral versions,themost natural one being the notion of a subsymmetric sequence or asubsymmetric set. In a normed space (X,· ), a sequence (xα)α∈I indexed inan ordered set (I,<) is called C-subsymmetricwhen
n j=1
ajxαj ≤C n j=1
ajxβj
foreverysequence of scalars(aj)nj=1 andeveryα1 <· · ·< αn and β1 <· · ·< βn inI.
WhenC= 1 thiscorrespondsexactlytothenotionofanindiscerniblesetanditiseasily seenthatthis canalways be assumed byrenorming X with an appropriate equivalent norm.Anothercloselyrelatednotionisunconditionality. Recallthatasequence(xi)i∈I
insomeBanachspace isC-unconditional whenever
i∈I
θiaixi ≤C
i∈I
aixi
for everysequence of scalars (ai)i∈I and everysequence (θi)i∈I of signs. Of particular interestaretheindiscerniblecoordinatesystems,suchastheSchauderbasicsequences.
The unit bases of the classical sequence spaces p, p ≥ 1 or c0 (in any density) are subsymmetricandunconditional(infact,symmetric,i.e.indiscerniblebypermutations) bases. Moreover, every basic sequence in one of these spaces has a symmetric subse- quence.Butthis isnottrueingeneral:therearebasicsequenceswithoutunconditional subsequences,thesimplestexamplebeing thesummingbasisof c0.However,itismore difficulttofindaweakly-nullbasiswithoutunconditionalsubsequences(B. Maureyand H.P. Rosenthal [17]).Nowweknowthatthere areBanachspaceswithoutunconditional
basicsequences. Thefirstsuch examplewasgivenby W.T. Gowersand B. Maurey [9], a spacewhichwasmoreoverreflexive.Concerningsubsymmetricsequences, wemention thattheunitbasisoftheSchreier space [23] doesnothavesubsymmetricsubsequences, andtheTsirelsonspace [27] isthefirstexampleofareflexivespacewithoutsubsymmetric basicsequences.
Allthese are separablespaces soit isnaturalto ask iflargespacesmust containin- finite unconditionalorsubsymmetricsequences,sincefromthetheoryoflargecardinals we know that infinite indiscernible sets exist in large structures. In general, this is a consequenceofcertainRamseyprinciples(i.e.higher-dimensionalversionsofthepigeon- holeprinciples).Indeed,itwasprovedbyKetonen[10] thatBanachspaceswithdensity biggerthanthefirstω-Erdőscardinalhavesubsymmetricsequences.Recallthatacardi- nal numberκiscalled ω-Erdőswheneverycountablecoloringofthecollectionoffinite subsetsofκhasaninfinitesubsetAofκwhere thecolorofagivenfinitesubsetF ofA dependsonlyonthecardinalityofF.Suchcardinals arelargecardinals,andtheirexis- tencecannotbe provedonthebasisofthestandardaxioms ofsettheory.Itistherefore naturaltoaskwhatistheminimalcardinalnumbernc(ns)suchthateveryBanachspace of densityat leastnc(resp.ns)hasanunconditional(respectively,subsymmetric) basic sequence. Itisnaturaltoconsider alsotherelative versionsofthesecardinals restricted to variousclassesofBanachspaceslike,forexample,theclassofreflexivespaceswhere we use thenotations ncrefl and nsrefl, respectively.Note thatit follows from E. Odell’s partial unconditionalityresult [21] thateveryweakly-null subsymmetricbasicsequence isunconditional,hencencrefl≤nsrefl.Moreover,aneasyapplicationofOdell’sresultand Rosenthal’s1-dichotomygivesthatnc≤ns.
Concerning lower boundsforthese cardinalnumbers,itwasprovedby S.A.Argyros and A. Tolias [3] that nc >2ℵ0, and by E. Odell [20] that ns>2ℵ0. For the reflexive case,weknowthatncrefl>ℵ1([5]),andintherecentpaper [1],S.A.ArgyrosandP. Mo- takis provedthatnsrefl >2ℵ0. Finally we mention thatin [16] the Erdőscardinals are characterized intermsoftheexistenceofcompactand largefamiliesandthesequential versionofnsrefl.MorepreciselyitisprovedthatthefirstErdőscardinaleωistheminimal cardinalκsuchthateverylongweakly-nullbasis oflengthκhasasubsymmetricbasic sequence,orequivalentlytheminimalcardinalκsuchthatthereisnocompactandlarge familyoffinite subsetsofκ.
Thestudyofupperboundsisofdifferentnatureandseemstoinvolvemoreadvanced set-theoretic considerations connected to large cardinal principles. This can be seen, for example, from the aforementioned result of Ketonen or from results of P. Dodos, J. Lopez-Abad and S. Todorcevic who proved in [7] that nc ≤ ℵω holds consistently relative to the existence of certain large cardinals and who proved in [16], based on classical results of A. Hajnal and P. Erdős [8] and R. M. Solovay [25], that Banach’s Lebesguemeasure extensionaxiomimpliesthatncrefl≤2ℵ0.
Inthispaperwecontinuetheresearchontheexistenceofsubsymmetricsequencesin anormed spaceof largedensity,and weprovethatnsrefl is ratherlarge,distinguishing thus the cardinals nsrefl and ncrefl. In contrast to the sequential version of nsrefl, that
is closely linked to indiscernibles of relational structures MF for compact and large familiesF, thefull version ofnsrefl ismorerelated to theexistenceof indiscerniblesin structuresthatarenotjustrelationalbutalsohaveoperations,suggestingthatnotonly we need to understandfamilies on finite sets but also “operations” with them.In the separable context, this is well-known and can be observed in the construction of the Tsirelson space, where finite products of the Schreier family are used in acrucial way (see[4], [6]). Thenaturalapproachinthe non-separablesetting wouldbe to generalize Tsirelson’s construction using analogues of the Schreier family, certain large compact families, on larger index sets. However, inthe uncountable level these families cannot be spreading and therefore, if one just copies Tsirelson’s construction on the basis of them,thecorresponding non-separable Tsirelson-likespaces will alwayscontain almost isometriccopiesof1([16,Theorem 8.2]).Thisleadsus tochangeourperspectiveanduse thewell-knowninterpolationtechnique [13,Example 3.b.10],anapproachthatappeared recentlyintheworkofArgyrosandMotakismentionedabove.Inthisperspective,a key tool is asuitableoperation ×, thatwe call multiplication, of compactfamilies offinite sets. In fact, the multiplication is an operation which associates to afamily F on the fixedindexset IandfamilyHonω,a familyF × HonI whichhas,inaprecisesense, manyelementsoftheform
n∈xsn,wherex∈ Hand(sn)n<ωisanarbitrarysequenceof elementsofF.Itiswellknownthatsuchmultiplicationexistsinωanditmodelsinsome waytheordinalmultiplicationonuniformfamilies.Itisalsothemain tooltodefinethe generalizedSchreierfamiliesonω,vastly usedinmodernBanachspace theory(see[4]) tostudyranksofcompactnotions(e.g.summabilitymethods),orofasymptoticnotions (e.g.spreadingmodels).Theseareuniformfamilies,sothatanyrestrictionofthemlooks like the entirefamily. Wegeneralize this property to the uncountablelevel bydefining homogeneousfamilies,thatdespitebeinguncountablefamiliesonlargeindex sets,have countableCantor–Bendixson rankwhich moreoverdoesnot changesubstantiallywhen passingtorestrictions.Inparticular,ifF ishomogeneous,thenthestructure MF does nothaveinfinitesets of indiscernibles,butwealso getlower and upperbounds forthe rankof thecollectionoftheir(finite)setsofindiscernibles.
We then introduce the notion of a basis of families, which is a rich collection of homogeneousfamiliesadmittingamultiplication, andweprovethattheyexistonquite largecardinalnumbers.Theexistenceofsuchbases isprovedinductively.Forexample, we prove that if κ has a basis then 2κ has also a basis. This is done by representing 2κ asthe completebinary tree2≤κ, andobserving thatwecanuse theheight function ht : 2≤κ→κ+1 topullbackabasisonκtoarestrictedversionofbasison2≤k,consisting of homogeneousfamilies of finite chains of2≤κ. Actually, we provethe following more generalequivalence(Theorem3.1).
Theorem. Foraninfinite rootedtreeT thefollowingare equivalent.
(a) There isabasisoffamilieson T.
(b) Thereis abasisoffamiliesconsisting of chainsof T and thereisabasisconsisting of antichainsof T.
In particular, we obtain a basis on 2ω that can be used to build a reflexive space of density2ω withoutsubsymmetricbasicsequences,givinganother proofof theresult in [1].Also,oneprovesinductivelythatforeverycardinalnumberκsmallerthanthefirst inaccessible cardinal,there isabasisonκandacorrespondingBanachspaceofdensity κ with similar properties. We then use Todorcevic’s methodof walks on ordinals [26]
to build trees on cardinals up to the first Mahlo cardinal number and find examples of reflexiveBanachspaces oflarge densitieswithoutsubsymmetric basicsubsequences.
Moreover,asobservedaboveforthestructureMF,wecanboundthecomplexityofthe (finite)subsymmetricbasicsequencesandobtainthefollowing.
Theorem. Every cardinal κ below the first Mahlo cardinal has a basis. Consequently, for every such cardinal κ and every α < ω1, there is a reflexive Banach space X of density κ with a long unconditional basis and such that every bounded sequence in X has anα1-spreadingmodelsubsequencebutthespaceXdoesnothaveβ1-spreadingmodel subsequences for β large enough, depending only on α. In particular, X contains no infinite subsymmetricbasic sequence.
The paperis organizedas follows.In Section2we introducesomebasictopological, combinatorialandalgebraic factsonfamiliesoffinite chainsofagivenpartialordering.
Wethendefinehomogeneousfamiliesandbasesofthem.Wepresentsomeupperbounds forthetopologicalrankofafamilythatusesthewell-knownRamseypropertyofbarriers on ω, a classical result by C. St. J. A. Nash-Williams ([19]), and that will be used several times along the paper. In Subsection 2.3 we give useful methods to transfer bases between partial orderings. Section 3 is the main part of this paper. The main object westudy is, givenatree, thecollectionAT C ofall finitesubtreesof T whose chains are in a fixed family C and such that the family of immediate successors of a given node is in another fixed family A. We study AT C both combinatorially and topologically. The combinatorial part is based on the canonical form of a sequence of finite subtrees, and allow us to define a natural multiplication. The topological one consists in finding upper bounds of the rank of the family AT C in terms of the correspondingranksofthefamiliesAandC,muchinthespiritofhowoneeasilybounds the size ofa finite tree from its height and splitting number.This operation allowsto lift bases on chains and of immediate successors to bases on the whole tree, our main result of this work done in Theorem 3.1. We apply this in Section 4 to prove that cardinal numbers smaller than the first Mahlo cardinal have a basis. To do this, we represent such cardinals as nodes of atree having bases on chainsand on immediate successors. We achieve this last part by proving several principles of transference of basis.Finally,weusebasestobuildreflexiveBanachspaceswithoutsubsymmetricbasic sequences.
Theauthorswishtothanktherefereeforhiscarefulreadingofthepaperandsugges- tionsthathaveimprovedtheexpositionoftheresults.
2. Basicdefinitions
Let I be a set. A set F is called a family on I when the elements of F are finite subsetsofI.LetP = (P,<) beapartialordering.AfamilyonchainsofP isafamilyon P consistingofchainsofP,i.e.totally orderedsubsetsofP.LetCh< bethecollection ofallfinite chainsofP.Given k≤ω,let
[I]k :={s⊆I : #s=k}, [P]k<:= [P]k∩Ch<, [I]≤k :={s⊆I : #s≤k}, [P]≤k< := [P]≤k∩Ch<.
ForafamilyF onI and A⊆I,letF A:=F ∩ P(A).RecallthatafamilyF onI is hereditary when it isclosed under subsetsand it iscompact when it is aclosed subset of 2I :={0,1}I, after identifyingeach set ofF with its characteristic function.In this case,F isascatteredcompactspace.SinceeachelementofF isfinite,itisnotdifficult tosee thatF is compactifand onlyifeverysequence (sn)n∈ω inF hasasubsequence (tn)n∈ωformingaΔ-systemwith rootinF,thatis,suchthat
tk0∩tk1=tl0∩tl1 ∈ F for everyk0=k1 andl0=l1.
The intersection tk∩tl, k = l is called the root of (tn)n. By weakening the notion of compactness,wesaythatF ispre-compact ifeverysequenceinF hasaΔ-subsequence (withrootnotnecessarilyinF).ItiseasytoseethatF ispre-compactifandonlyifits
⊆-closure {s⊆I : s⊆t for somet∈ F}iscompact.
RecalltheCantor–BendixsonderivativesofatopologicalspaceX: X(0):=X, X(α)=
β<α
(X(β))
whereY denotesthecollectionofaccumulationpointsofY,thatis,thosepointsp∈X suchthateachofitsopen neighborhoodshasinfinitelymanypointsinY.Theminimal ordinalαsuchthatX(α+1)=X(α)iscalledtheCantor–BendixsonrankrkCB(X) ofX.In thecaseofacompactfamilyF onanindexsetI,beingscattered,itsCantor–Bendixson indexisthefirstαsuchthatF(α)=∅,andthereforeitmustbe asuccessor ordinal.
Definition2.1(Rank,smallrankandhomogeneousfamilies).GivenacompactfamilyF onsomeindex setI,let
rk(F) := rkCB(F)−
where (α+ 1)−=α.We saythatacompactfamily F iscountably rankedwhen rk(F) iscountable.LetP beapartialordering.ForagivenfamilyF onchainsofP,thesmall rankrelative toP ofF is
srkP(F) := inf{rk(F C) : C is an infinite chain of P}.
A compact and hereditary family F on chains of P is called (α,P)-homogeneous if F ={∅}ifα= 0,and if1≤α < ω1, then[P]≤1⊆ F and
α= srkP(F)≤rk(F)< ι(α),
whereι(α) isdefinedbelowinDefinition2.4.FisP-homogeneousifitis(α,P)-homogene- ousforsomeα < ω1.
Observe thatthese notions coincidefor any two total orders onthe fixed index set, as being a chain with respect to atotal order does not depend on the total order it- self. Hence, when F is a familyon some index set I, we use srk, α-homogeneous and homogeneous for the corresponding notionswith respect to any total order onI. If F is acompact familyonacountable indexset I, thenrk(F) is countableand,therefore, the small rank of F with respect to any partial order on I is countable. In general,
#rk(F)≤#I andtheextremecasecanbeachieved.
Definition 2.2.The normal Cantor form of an ordinal αis the unique expression α = ωα[0]·n0[α]+· · ·+ωα[k]·nk[α] where α≥α[0]> α[1]>· · ·> α[k]≥0 andni[α]< ω foreveryi≤k.
Suppose that∗ is an operation on countable ordinals and suppose that α > 0 is a countableordinal.Wesaythatαis∗-indecomposablewhenβ∗γ < αforeveryβ,γ < α.
Remark2.3.It is well-knownthat
(i) αissum-indecomposableifand onlyifα=ωβ.
(ii) α > 1 is product-indecomposable if and only if α = ωβ for some sum-indecom- posableβ.
(iii) Forα > ω,αisexponential-indecomposableifandonlyifα=ωα.
(iv) Product-indecomposabilityimplysum-indecomposability,andexponential-indecom- posabilityimplyproductand sumindecomposability.
So,1,ω,ω2and1,ω,ωωarethefirst 3sum-indecomposable,andproduct-indecompos- able ordinals, respectively. If we define, given α < ω1, α0 := α, αn+1 := ωαn and αω := supnαn, then ω,ωω,(ωω)ω are the first 3 exponential-indecomposableordinals.
Wewill useexp-indecomposabletorefertoexponential-indecomposableordinals.
Definition2.4.Given acountableordinalα,let
ι(α) = min{λ > α : λis exp-indecomposable}.
LetFn(ω1,ω) bethecollectionofallfunctionsf :ω1→ωsuchthatsuppf :={γ < ω1 : f(γ)= 0} is finite. Endowed with the pointwise sum +, (Fn(ω1,ω),+) is an ordered commutative monoid. Let ν : ω1 → Fn(ω1,ω) be defined by ν(α)(γ) = ni[α] if and onlyif γ=α[i]. Letσ: Fn(ω1,ω)→ω1 be definedbyσ(f)=
i≤nωαi ·f(αi),where {α0 >· · ·> αn ≥0}= suppf. Inother words, σis theinverse ofν.Given α,β < ω1, theHessenbergsum (seee.g. [24])isdefinedby
α+β˙ :=σ(ν(α) +ν(β)).
Itiseasytosee thatifαisexp-indecomposable,thenitis+-indecomposable.˙ Definition2.5.LetF andGbe familiesonchainsofapartialorderingP.Define
F ∪ G:={s⊆P : s∈ F ors∈ G},
F PG:={s∪t : s∪tis a chain and s∈ F, t∈ G}, F G:={s∪t : s∈ F, t∈ G},
FP(n+ 1) := (FP n)PF; FP1 :=F, F(n+ 1) := (Fn) F; F1 :=F.
ObservethatwhenP isatotalorderingtheoperationsP and arethesame.
Proposition 2.6.The operations∪, P and preserve pre-compactness andhereditari- ness.Moreover, if F andG are countablyranked familieson chainsof P,then
(i) rk(F ∪ G)= max{rk(F),rk(G)}, (ii) rk(F G)= rk(F) ˙+rk(G), (iii) rk(F PG)≤rk(F) ˙+rk(G).
Consequently, if F and G are P-homogeneous, then F ∪ G, F P G and F G are (γ,P)-homogeneous with γ≥max{srkP(F),srkP(G)}.
Proof. It is easy to see that if F and G are pre-compact, hereditary, then F ∗ G is pre-compact,hereditary,for∗∈ {∪,P,}. Letussee (i):Aneasyinductive argument showsthat(F ∪ G)(α)=F(α)∪ G(α)foreverycountableα.Toprove (ii)and (iii),notice thatbyageneralfact,foreverycompactspacesK andLandeveryαonehasthat
(K×L)(α)=
β+γ=α˙
(K(β)×L(γ)). (1)
When rk(K) and rk(L) are countable, we have that rk(K×L) = rk(K) ˙+rk(L). The proof of (1) is done by induction on α and by considering the case when α is sum- indecomposableornot.NowletFandGbecountablyranked.SupposethatP isatotal ordering,andletF ×G → F G,(s,t)→s∪t.Thisisclearlycontinuous,ontoandfinite- to-one,sork(F G)= rk(F × G)= rk(F) ˙+rk(G),whichconcludestheproofof(ii).IfP isingeneralapartialordering,thenitfollowsfromthisthatrk(F PG)≤rk(F) ˙+rk(G), proving (iii).Nowsuppose thatF and GareP-homogeneous.Wehaveclearlythat
max{srkP(F),srkP(G)} ≤min{srkP(F ∪ G),srkP(F PG),srkP(F G)}. (2) On the other hand, max{rk(F ∪ G),rk(F P G),rk(F G)} ≤ rk(F) ˙+rk(G). Since rk(F),rk(G) < λ := max{ι(srkP(F)),ι(srkP(G)), it follows by the indecomposability of λthatrk(F) ˙+rk(G)< λ.This,togetherwith (2) givesthedesiredresult. 2
2.1. Basesof homogeneousfamilies
Werecallawell-knowngeneralizationofSchreierfamiliesonω,calleduniformfamilies.
Wearegoingtousethemmainlyasatooltocomputeupperboundsofranksofoperations ofcompactfamilies.Weusethefollowingstandardnotation:givenM,s,t⊆ω,wewrite s < t to denote thatmaxs <mint and letM/s :={m∈M : s < m}. Notice thata familyFonωispre-compactifandonlyifeverysequenceinFhasablockΔ-subsequence (sn)n∈ω, that is, such that s < sm\s < sn\s for every m < n, where s is the root of (sn)n. We write s t to denote that s is an initial part of t, that is, s ⊆ t and t∩(maxs+ 1)=s,ands<t todenotethatstands=t.
Definition 2.7.GivenafamilyF onω andn< ω,let
F{n}:={s⊆ω : n < sand{n} ∪s∈ F}.
Letαbeacountableordinalnumber,andletF beafamilyonaninfinitesubsetM⊆ω.
ThefamilyF is calledanα-uniformfamily onM when ∅∈ F and (a) F ={∅}ifα= 0;
(b) F{n} isβ-uniformonM/nforeveryn∈M,ifα=β+ 1;
(c) F{n}isαn-uniformonM/nforeveryn∈M and(αn)n∈M isanincreasingsequence suchthatsupn∈Mαn=α,ifαislimit.
Itisimportanttoremarkthatuniformfamiliesarenotuniformfronts,whichwerein- troducedbyP.PudlakandV.Rödlin [22] followingpreviousworksofC.Nash-Williams.
Recall thatafamilyBonM iscalled anα-uniformfrontonM whenB={∅}ifα= 0, and if α > 0 then ∅ ∈ B/ , and B{n} is a γ-uniform front on M/n for every n ∈ M, if β =γ+ 1, and B{n} is aαn-uniform fronton M/n for every n ∈ M and (αn)n∈M
is increasing with supn∈Mαn. In fact, given a uniform family F, the collection of its
⊆-maximal elements Fmax isauniform front,andwe canrecover auniformfamilyout ofauniform frontBbytaking itsclosure underinitial parts B, thatis, thecollection ofinitialparts ofelementsofB:
Proposition2.8.
(a) Everyuniform familyiscompact.
(b) Thefollowingare equivalent:
(b.1) F isan α-uniformfamily onM.
(b.2) Fmax isan α-uniform front onM suchthat F=Fmax= (Fmax).
Proof. (a) is provedbya simpleinductiveargument on α. To provethat(b.1)implies (b.2),oneobserves first that(Fmax) =F, becauseF iscompact, and thenagain use an inductive argument. The proof of that (b.2)implies (b.1) one uses the well-known factthatifBisauniformfront,thenB=B (seeforexample [2]). 2
Definition2.9.Given twofamiliesF andG onω theirsumand productaredefinedby F ⊕ G:={s∪t : s < t, s∈ G and t∈ F},
F ⊗ G:={
i<n
si : {si}i⊆ F, maxsi<minsi+1, i < n, and{minsi}i∈ G}. Thefollowingarewell-knownfactsofuniformfronts,andthatareextendedtouniform familiesbyusing thepreviousproposition.Formoreinformation onuniform fronts,we referto [14],[15] and [2].
Proposition2.10.
(a) The rankof anα-uniformfamily isα.
(b) Theunique n-uniform family onM,n< ω,is[M]≤n.
(c) If F is uniform and t is a finite set, the Ft := {s⊆ω : t < s andt∪s∈ F} is uniform onω/t.
(d) If F is an α-uniform family on M,then F N is an α-uniform family on N for every N ⊆M infinite. Consequently, if F is an α-uniform family on ω, then F is α-homogeneous withsrk(F)= rk(F)=α.
(e) If F isanα-uniformfamilyonM,andθ:M→N isanorder-preservingbijection, then {θ(s) : s∈ F}isan α-uniform family onN.
(f) SupposethatF andG areαandβ uniformfamiliesonM,respectively.ThenF ∪G, F ⊕ G, F G and F ⊗ G are max{α,β}, α+β, α+β˙ and α·β-uniform families on M,respectively.
(g) Uniform fronts have the Ramseyproperty: if c:F →n is acoloringof a uniform front on M,thenthereis N⊆M infinitesuch thatc isconstant on FN.
(h) SupposethatF andG areuniformfamiliesonM.ThenthereissomeN ⊆M such that either F N ⊆ G or G N ⊆ F. Moreover, when rk(F) < rk(G), the first alternative must holdandinaddition (FN)max∩(GN)max=∅.
(i) If F is a uniform family on M, then there isN ⊆M infinite such that F N is hereditary.
(j) If F is compact and -hereditary family on ω, then there is M ⊆ ω infinite such that FM isauniform familyon M. 2
Remark2.11.
(i) The onlynew observationin theprevious propositionis the factin(f) thatstates thatunionsandsquareunionsofuniformfamiliesisauniformfamily,andthatcan beeasilyprovedbyinductiononthemaximumoftheranksusing,forexample,that (F ⊗ G){n}= (F ⊗ G{n})⊕ F{n}.
(ii) A simple inductive argument shows that for every countable α and every infinite M ⊆ω there is an α-uniform family F on M, and, although uniform familiesare notnecessarilyhereditary using(d)and(i)onecanbuildthem beinghereditary.
Weobtainthefollowingconsequenceforfamiliesonanarbitrarypartialordering.
Corollary 2.12.SupposethatP = (P,<P)isapartial ordering,andsupposethat F and G arecompactandhereditaryfamilieswithrk(F)<srk(G).Theneveryinfinitechain C of P has aninfinite subchainD⊆C suchthat FD⊆ G.
Proof. Fix F, G and C as in the statement. By going to a subchain of C, we may assumethatC iscountable.Since (C,<P),so wecanfix abijection θ:C→ω,andwe set F0:={θ(s) : s∈ F C}and G0:={θ(s) : s∈ GC}.ObservethatG0 M has rank at least srk(G) >rk(F0) for everyinfinite M ⊆ω. Then by Proposition2.10 (j), (h)there issomeinfiniteM ⊆ωsuchthatF0M⊆ G0,soF θ−1(M)⊆ G. 2
Among uniform families, the generalizedSchreier familieshave been widely studied andusedparticularlyinBanachspacetheory.Theyhaveanalgebraicdefinitionandhave extraproperties,asforexamplebeingspreading.Also,theyhaveasum-indecomposable rank.Werecallthedefinitionnow.
Definition 2.13.TheSchreier family is
S :={s⊆ω : #s≤mins}. A Schreiersequence isdefinedinductivelyforα < ω1 by (a) S0:= [ω]≤1,
(b) Sα+1:=Sα⊗ S,and
(c) Sα:=
n<ω(Sαnω\n) where(αn)n is suchthatsupnαn=α,ifαislimit.
Note thatthefamily Sα depends onthe choiceof thesequences convergingto limit ordinals.
Definition 2.14(Spreading families). Afamily F onω is spreading when forevery s= {m0<· · ·< mk}∈ F and t ={n0 <· · ·< nk}with mi ≤ni for everyi≤k onehas thatt∈ F.
Proposition2.15.
(a) Suppose that F andG are spreading familiesonω. ThenF ∪ G,F ⊕ G andF ⊗ G are spreading. Ifin additionF orG ishereditary, thenF G isalso spreading.
(b) If F iscompact, hereditaryandspreading, thenF is rk(F)-homogeneous.
(c) SupposethatF isacompact,hereditaryandspreadingfamilyonω offiniterankm.
Then thereissomen< ω suchthat[ω\n]≤m⊆ F ⊆[ω]≤m.
Proof. (a)is easytoverify.For(b):Given aninfinitesetM ⊆ω,letθ:ω →M be the uniqueincreasing enumeration of M. Then s →θs is a 1–1and continuous mapping from F into F M, so rk(F M)≥rk(F)≥ rk(F M). (c): If there is some s∈ F of cardinality at least m+ 1, then [ω\maxs]≤m+1 ⊆ F because F is hereditary and spreading.Andthisisimpossible,asitimpliesthattherankofFisatleastm+1.Onthe otherhand,sincetherankofF ism,theremustbesomeelementofitofcardinalitym.
Pickanelement sofF ofsuchcardinality.Then[ω\maxs]≤m⊆ F. 2
ThegeneralizedSchreierfamiliesareuniformfamiliesandtheyhaveextraproperties.
Proposition2.16.
(a) Sα is hereditary,spreadingand ωα-uniform.
(b) Foreveryα≤β there isn< ω suchthat Sα(ω\n)⊆ Sβ.
Proof. (a)Thefirsttwopropertiesarewell-known.TheproofofthatSαisaωα-uniform family is done by induction on α. The case α = 0 is trivial,while it is easy to verify thatS isanω-uniform family,soSα+1 =Sα⊗ S isaωα·ω=ωα+1-uniform familyby Proposition2.10andinductivehypothesis.Supposethatαislimit.Foragivenm,n∈ω, letαnm< ωαn be suchthat(αnm)m isincreasing, supmαnm =αn and (Sαn){m} is aαnm uniformfamily.Sinceforeverym∈ω wehavethat
(Sα){m}=
n≤m
(Sαn){m} (ω\m)
itfollowsfromProposition2.10(e)that(Sα){m}isaβm:= maxn≤mαnm-uniformfamily onω/m.Itiseasytoseethat(βm)misincreasingandsatisfiesthatsupmβm=ωα.(b) is provedbyasimpleinductiveargument. 2
Definition2.17.LetSbethecollectionofallhereditary,spreadinguniformfamiliesonω.
Proposition 2.18.For every α < ω1 there is a hereditary, spreading α-uniform family on ω.
Proof. Let(Sα)α<ωbeaSchreiersequence,andgivenacountableordinalαwithnormal Cantor formα=
i≤kωαi·ni wedefine
Fα:= (Sα0⊗[ω]≤n0)⊕ · · · ⊕(Sαk⊗[ω]≤nk).
Then eachFα isahereditary,spreadingα-uniformfamilyonω. 2
Wepresent nowtheconceptofbasis,whichisacollectionof familiesthatintendsto generalize the collectionofuniform family onω,and themultiplication ⊗between the familiesinthebasis.Itseemsthatthereisnocanonicaldefinitionforthemultiplication F × G oftwofamiliesonanindexsetI.However,whenGisafamilyonωwecandefine itquitenaturallyasfollows.
Definition 2.19.LetF beahomogeneousfamily onchainsofapartial orderingP, and let H be a homogeneous family on ω. We say that a family G on chains of P is a multiplication of F byHwhen
(M.1) G ishomogeneousandι(srkP(G))=ι(srkP(F)·srk(H)).
(M.2) Every sequence (sn)n<ω in F such that
nsn is a chain of P has an infinite subsequence(tn)n suchthatforeveryx∈ Honehasthat
n∈xtn∈ G. Example 2.20.
(i) F P n is a multiplication of any homogeneous family F by [ω]≤n. In general, given afamily H ∈S offinite rank n, FP n is amultiplication of F by H(see Proposition2.15(b)).
(ii) IfP doesnothaveany infinitechain,then anyhomogeneousfamilyF onchainsof P hasfiniterankandgivenanyhomogeneousfamilyHonω,G=F satisfies(M.2).
Notice that always F ⊆ G for every multiplication G of F by any family H = {∅}, and that when F,H = {∅}, then (M.1) is equivalent to ι(srkP(G)) = max{ι(srkP(F)),ι(srk(H))}, becauseι(α·β)= max{ι(α),ι(β)}if α,β ≥1. When the family F = [κ]≤1 and H is the Schreier family S, then the existence of a family G satisfying (M.2) is equivalent to κ not being ω-Erdős (see [16], and the remarks after
Theorem 2.23). Let us use the following notation. Given a collection C of families on chainsofP andα < ω1 letCα:={F ∈C : srkP(F) =α}.
Definition 2.21 (Basis).Let P = (P,<) be a partial ordering with an infinite chain.
Abasis(of homogeneousfamilies)on chainsof P isapair(B,×) suchthat:
(B.1) B consists of homogeneous familieson chains of P, it contains all cubes [P]≤Pn, andBα=∅forallω≤α < ω1.
(B.2) Bis closed under∪ and P, and ifF ⊆ G ∈Bis homogeneous onchainsof P andι(srkP(F))=ι(srkP(G)),thenF ∈B.
(B.3) × : B×S → Bis such thatfor every F ∈ Band every H ∈ S onehas that F × H isamultiplication ofF byH.
WhenP= (P,<) isatotalordering,wesimplysaythatBisabasis offamilieson P.
MultiplicationsF × HforfamiliesHoffiniterankalwaysexist,so(B.3)isequivalent totheexistenceofmultiplicationbyfamiliesH ∈Sofinfiniterank.Also,{0}×H={∅}
isalwaysamultiplication. Theconditionon(B.2) concerningsubfamilies isatechnical requirement,butisnotessential(see Proposition2.24).
Proposition2.22.There isbasisoffamilieson ω.
Proof. Let Bbe thecollection of all homogeneous familiesF such that there is some uniform family G with F ⊆ G and ι(srk(F)) = ι(srk(G)). Given {∅} F ∈ B and H ∈S,choosesomeuniformfamilyGwithF ⊆ G andι(srk(F))=ι(srk(G)),anddefine
F ×ωH:= (G ⊗ H)⊕ G.
(B.1): Bα = ∅ follows from Proposition 2.18. (B.2): B is closed under ∪ and by Proposition 2.6 and Proposition 2.10. (B.3): Fix {∅} F ∈ B, and H ∈ G.
Choose an homogeneous family G defining F ×ωH = (G ⊗ H)⊕ G. We verify first (M.1): Let α,β < ω1 be such that G and H are α-uniform and β-uniform, respec- tively. Observe that ι(skr(F)) = ι(srk(G)) = ι(α) and srk(H) = β. Then F ×ωH is α·(β+ 1)-uniform, srk(F ×ωH) =α(β+ 1), so ι(srk(F ×ωH)) = max{ι(α),ι(β)}= max{ι(srk(F)),ι(srk(H))} = ι(srk(F)·srk(H)). This finishes the proof of (M.1). We checknow(M.2):Supposethat(sk)kisasequenceinF.Let(tk)k<ωbeaΔ-subsequence with root t ∈ F such that t < tk \t < tk+1 \t for every k. Suppose that x ∈ H. Then{mintk\t}k∈x∈ H, becauseHisspreading. Hence,
k∈x(tk\t)∈ G ⊗ H. Since t<
k∈x(tk\t),itfollowsthat
k∈xtk=t∪
k∈x(tk\t)∈(G ⊗ H)⊕ G. 2
Ourmain resultisthefollowing:
Theorem2.23.Everycardinalθ strictlysmaller thanthefirstMahlo cardinalhasabasis of familieson θ.
Theproof,inSection4,isdoneinductivelyonκandusingtreeswithsmallheightand levels(tobeprecisedlater)inordertostepup.Forexample,whenκisnotstronglimit, there mustbe λ< κ suchthat2λ≥κ,so,bytheinductivehypothesis,theremust bea basisonλ,and thereisarathernaturalwayto liftituptoabasisonthenodes ofthe complete binarytree,viatheheightmapping.Thecasewhenκisnotregularissimilar.
Whenonthecontraryκisinaccessible, thetreeT ofcardinalityκissubstantiallymore complicated,andinfactreliesonthemethodofwalksinordinals.Inanyofthesecases, the main difficultyisto passfrom abasis onchainsand abasis“on theantichains” of a treeT to abasisof familiesonthe nodes ofT withrespect to any total ordering on them,thusgettingabasison|T|(seeTheorem 3.1).
Recall that κis ω-Erdős when for every coloring c : [κ]<ω → 2 there is an infinite c-homogeneoussubsetA⊆κ,thatis,fors,t∈A,c(s)=c(t) when#s= #t.A compact and hereditary family on κis called large when srk(F) ≥ω, or, equivalently, when F satisfies(M.2)for[κ]≤1 andtheSchreierfamilyS.Itisprovedin[16] thattheexistence of suchfamiliesinκisequivalent toκnotbeing ω-Erdős.
Problem 1.Characterizewhenκhas≥ω-homogeneousfamilies.
Problem 2.Characterizethe cardinal numbers κ such that there exists c : [κ]<ω → 2 withoutinfinitec-homogeneoussets,butsuchthatthereisa≥ω-homogeneousfamilyF onκsuchthatforeverysequence(sn)n<ω inF and everyl < ωthere aren1<· · ·< nl
suchthatl
i=1sni isc-homogeneous.
ThefirstsuchκnotsatisfyingthiscoloringpropertyisatleastthefirstMahlocardinal and smallerthanthefirstω-Erdőscardinal.
Before goinginto theparticular caseofthepartial ordering being atree, wepresent someoperationsofpartial orderingsandresultsguaranteeingthatwecantransferfam- ilies or bases from somepartial orderings to morecomplex ones (Subsection 2.3). The following characterizationoftheexistenceofabasiswillbeuseful.
Proposition 2.24. A partial ordering P with an infinite chain has a basisif and only if there isapair(B,×),called pseudo-basis,with thefollowingproperties:
(B.1) Bconsistsofhomogeneousfamiliesonchainsof P,itcontainsallcubes,andfor every ω≤α < ω1 there isF ∈Bsuchthat α≤srkP(F)≤ι(α).
(B.2) Bisclosed under ∪andP.
(B.3) ×:B×S≥ω→B,S≥ω being theinfinite ranked familiesof S,issuchthat for every F ∈BandeveryH ∈Sonehas thatF × HisamultiplicationofF byH.
Proof. Suppose that (B,×) satisfies (B.1), (B.2) and (B.3). Let C = {pn}n be an infinite chainof P, of order typeω. Fix abasis (B(ω),×ω) of families onω. For each G ∈B(ω), letG :={{pn}n∈x : x∈ G}. Then G ishomeomorphic to G.Given F ∈B, letF:={s∈ F : s∩C=∅}.NowletB bethecollectionofallunionsF ∪ G suchthat F ∈B,G ∈S,andfinallyletBbethecollectionofallP-homogeneousfamiliesF such thatthereis someG ∈B withF ⊆ G and ι(srkP(F))=ι(srkP(G)). ForeachF ∈B we chooseGF ∈ Band HF ∈ B(ω) suchthat F ⊆GF∪ HF ∈ B and ι(srkP(F))= ι(srkP(GF∪ HF)).IfH hasinfiniterank,wedefine F ×H:= (GF× H)∪(HF×ωH), andwhen Hhasfinite rankn,letF ×H:=FP n.It iseasyto check that(B,×) isabasisonchainsofP. 2
2.2. Ranks anduniformfamilies
Theobjectiveofthispartistogivetechnicaltoolsforgettingupperboundsforranks withtheuseofuniformfamilies.
Definition 2.25.Given families F on ω and G on a partial ordering P, we say that a mappingf : F → G betweentwo families is(,⊆)-increasing whens t impliesthat f(s)⊆f(t).
Thefactthatapointis inacertainderivativeofacompactmetrizable spaceK can bewitnessedbyacontinuousand 1–1mappingfromauniformfamilyintoK.
Proposition 2.26 (Parametrization of ranks).Suppose that K is a compact metrizable space and let α < ω1. Then a point p ∈ K is such that p ∈ K(α) if and only if for every α-uniform family B there is a1–1 and continuous functionθ :B →K suchthat θ(∅)= p. In case K = F is a compact family on I, p∈ F(α) if and only if for every α-uniformfamilyBthereisa1–1andcontinuousmappingθ:B → F suchthatp=θ(∅) andsuchthat θ is(,⊆)-increasing.
Proof. Given p∈K andε>0,letB(p,ε) betheopenball aroundpandradius ε.The proofis byinductiononα.Suppose thatp∈K(α) andletB be aα-uniform familyon MandCthecollectionof-maximalsubsetsinB.Withoutlossofgeneralityweassume that M = ω. Let αn < α be such that C{n} is αn-uniform on ω/n. Choose (pn)n in K(αn)convergingnon-triviallytopsuchthattherearemutuallydisjointclosedballsBn
aroundpnwithdiam(Bn)↓n 0.Sinceeachpn∈K(αn)itfollowsbyinductivehypothesis thatforeachnthereis a1–1andcontinuousfunction
θn :B{n}=C{n}→Bn
withθn(∅)=pn.Letθ:B →K be definedbyθ(∅)=p, θ(s):=θmins(s\ {mins}), for s =∅. By the choiceof theballs Bn it follows thatθ is 1–1. Weverify now thatθ is
