Uniforming n-place functions on ds(alpha)

0.22 MB·

by  Esther Gruenhut

Uniforming n-place functions on T ⊆ds(α) 1
In this paper the Erdo˝s-Rado theorem is generalized to the class of well founded
trees. We define anequivalence relationonthe class ds(∞)<ℵ0 (finitesequences of
decreasingsequencesofordinals)withℵ0 equivalenceclasses,andforn<ωanotion
ofn-end-uniformityforacolouringofds(∞)<ℵ0 withµcolours.Wethenshowthat
for every ordinal α, n < ω and cardinal µ there is an ordinal λ so that for any
8 colouring c of T = ds(λ)<ℵ0 with µ colours, T contains S isomorphic to ds(α) so
0 that c↾S<ℵ0 is n-end uniform. For c with domain Tn this is equivalent to finding
0 S ⊆ T isomorphic to ds(α) so that c↾Sn depends only on the equivalence class of
2 thedefinedrelation,soinparticularT →(ds(α))n .Wealsodrawaconclusionon
n colouringsofn-tuplesfromascattered linearordµer,ℵ.0
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Combinatorics, Probability and Computing(2008)00,000–000. (cid:13)c 2008CambridgeUniversityPress
DOI:10.1017/S0000000000000000 PrintedintheUnitedKingdom
n T α
Uniforming -place functions on ⊆ ds( )
E. GRUENHUT1 and S. SHELAH2†
1 EinsteinInstituteofMathematics, TheHebrewUniversityofJerusalem,
Jerusalem91904, Israel.
2 EinsteinInstituteofMathematics, TheHebrewUniversityofJerusalem,
Jerusalem91904, Israel,
andDepartmentofMathematics,RutgersUniversity,
NewBrunswick,NJ08854,USA.
[email protected]@math.huji.ac.i
This paper is a natural continuation of [3] in which Shelah and Komja´th prove that
for any scattered order type ϕ and cardinal µ there exists a scattered order type ψ such
that ψ →[ϕ]n . This was proved by a theorem on colourings of well founded trees. By
µ,ℵ0
Hausdorff’scharacterization(see[2]and[4])everyscatteredordertypecanbeembedded
in a wellfounded tree,so we candeduce a natural generalizationof their theoremto the
n-arycase,i.eforeveryscatteredordertypeϕ,n<ω,andcardinalµthereisascattered
order type ψ such that ψ →(ϕ)n .
µ,ℵ0
We start with a few definitions.
Definition 0.1. For an ordinal α we define ds(α) = {η : η a decreasing sequence of
ordinals <α}. By ds(∞) we mean the class of decreasing sequences of ordinals.
We sayT ⊆ds(∞)isatreewhenT isnon-emptyandclosedunderinitialsegments.T,S
will denote trees. For S ⊆T ⊆ds(∞) we say that S is a subtree of T if it is also a tree.
We use the following notation:
Notation 0.2.
1 For η,ν ∈ds(∞) by η∩ν we mean η↾ℓ where ℓ is maximal such that η↾ℓ=ν↾ℓ.
2 For η ∈ds(∞) and a tree T ⊂ds(∞) we define
η⌢T ={ρ:ρEη∨(∃ν ∈T)(ρ=η⌢ν)}
† ResearchofbothauthorssupportedbytheUnitedStates-IsraelBinationalScienceFoundation(Grant
no.2002323). Publication909inShelah’sarchive.
Uniforming n-place functions on T ⊆ds(α) 3
Notethatforη ∈ds(∞\{hi})and{hi}(T ⊆ds(∞)ifη(lg(η)−1)>sup{ρ(0):ρ∈T}
then η⌢T ⊆ds(∞).
Definition 0.3. We define the following four binary relations on ds(∞):
1 Let <1 be the two place relation on ds(∞) defined by η <1 ν iff one of the following:
ℓx ℓx
(∃ℓ)(η(ℓ)<ν(ℓ) or η↾ℓ=ν↾ℓ) or η⊳ν.
2 Let <2 be the two place relation on ds(∞) defined by η <2 ν iff one of the following:
ℓx ℓx
(∃ℓ)(η(ℓ)<ν(ℓ) or η↾ℓ=ν↾ℓ) or ν⊳η.
3 <∗ =<1 ∩<2 .
ℓx ℓx ℓx
4 Let <3 be the two place relation on ds(∞) defined by η <3 ν iff one of the following:
η⊳ν or for the maximal ℓ such that η↾ℓ = ν↾ℓ if ℓ is even then η(ℓ) < ν(ℓ) and if ℓ
is odd then η(ℓ)>ν(ℓ).
Itiseasilyverifiedthat<1 ,<2 and<3 arecompleteordersofds(∞),andtherefore<∗
ℓx ℓx ℓx
is a partial order. The following remark refers to their order types defined by <1 ,<2
ℓx ℓx
and <3 on ds(∞) or ds(α).
Observation 0.4.
1 <1 ,<2 are well orderings for ds(∞).
ℓx ℓx
2 (ds(α),<3) is a scattered linear order type for every ordinal α.
3 Every scattered linear order type can be embedded in (ds(α),<3) for some ordinal α.
Proof.
1 Let ∅6=A⊆ds(∞), we define by induction on n<ω an element a in the following
n
manner a = min{η(0) : η ∈ A}, assume a ,··· ,a have been chosen so that
0 0 n−1
ha : k < ni ∈ ds(∞) and for every η ∈ A ha : k < ni ≤2 η↾n (if lg(η) ≤ n then
k k ℓx
η↾n = η). Now choose a = min{η(n) : η ∈ A∧η↾n = ha : k < ni}, if that set
n k
isn’tempty. As the sequence derivedin the abovemanner is a decreasingsequenceof
ordinals it is finite, say a ,···a have been defined and a cannot be defined, we
0 n−1 n
will show that a¯=ha :k <ni is the minimal element of A with respect to <2 . By
k ℓx
the definition of the sequence there is an η ∈A so that η↾n=a¯, if lg(η)>n then we
could have defined a , so η =a¯ and in particular a¯∈A, and for every η ∈A\{a¯} we
n
have a¯<2 η. Let n =min{m:a¯↾m∈A} so a¯↾n is the minimal <1 element in A.
ℓx ∗ ∗ ℓx
2 The proof is by induction on α. Assume that (ds(β),<3) is a scattered linear order
type for every β < α, and assume towards contradiction that Q can be embedded
in (ds(α),<3), q 7→ ηq. Let C = {ℓ : (∃p,q ∈ Q)(ηp(ℓ) 6= ηq(ℓ))}, ℓ = minC and
Γ = {β : (∃q ∈ Q)(ηq(ℓ) = β)}. Without loss of generality ℓ is even and for β0 =
minΓ, β1 = minΓ\{β0} there are q0 < q1 ∈ Q so that ηqi(ℓ) = βi, i = 0,1. Now
(q ,q ) = B ∪B where B = {p ∈ (q ,q ) : η (ℓ) = β }. For some i ∈ {0,1} the
0 1 0 1 i 0 1 p i
set Bi contains an interval of Q and is embedded in (ηqi↾(ℓ+1)⌢ds(βi),<3) but this
would imply that Q can be embedded in (ds(βi),<3) which is a contradiction to the
induction hypothesis.
3 By Hausdorff’s characterization it is enough to show for ordinals α and β that both
4 E. Gruenhut and S. Shelah
A = (ds(α),<3)×β and A = (ds(α),<3)×β∗ can be embedded in (ds(α+
α,β α,β∗
β·2+1),<3). The embedding is given as follows, for (η,γ)∈A we have (η,γ)7→
α,β
hα+β+γ+1,α+βi⌢η,andfor(η,γ)∈A wehave(η,γ)7→hα+β·2,α+β+γi⌢η.
α,β∗
Definition 0.5. For trees T ,T ⊂ ds(∞), f : T → T is an embedding of T into T
1 2 1 2 1 2
if f preserves level, ⊳ and <1 (or equivalently, <2 ,<∗ or <3).
ℓx ℓx ℓx
Observation 0.6. For trees T ,T ⊂ds(∞), if f :T →T preserves level and ⊳ then
1 2 1 2
in order to determine whether f is an embedding it is enough to check for η ∈ T and
1
ordinals γ <γ such that ν =η⌢hγ i∈T (i=1,2) that f(ν )<∗ f(ν ).
1 2 i i 1 1 ℓx 2
As T ⊆ds(∞) is well founded, i.e there are no infinite branches,it is natural to define a
rank function. in the following definition rk isn’t the standard rank function but for
T,µ
µ=1wegetasimilardefinitiontothe usualdefinitionofarankonawellfoundedtree.
Definition 0.7. For a tree T ⊂ ds(∞) and cardinal µ define rk (η) : ds(∞) →
T,µ
{−1}∪Ord by induction on α as follows:
(a) rk (η)≥0 iff η ∈T.
T,µ
(b) rk (η)≥α+1 iff µ≤|{γ :η⌢hγi∈T ∧rk (η⌢hγi)≥α}|.
T,µ T,µ
(c) rk (η)≥δ limit iff (∀α<δ)(rk (η)≥α).
T,µ T,µ
We say that rkT,µ(η)=α iff rkT,µ(η)≥α but rkT,µ(η)(cid:3)α+1.
Denote rk (T)=rk (hi), and rk (η)=rk (η).
T,µ T,µ T T,1
Definition 0.8. For a tree T ⊂ds(∞), η ∈T and cardinals µ,λ we define the reduced
rank rkλ (η)=min{λ,rk (η)}.
T,µ T,µ
We first note a few properties of the rank function.
Observation 0.9. For η ∈T ⊂ds(∞) and an ordinal α we have:
1 For cardinals µ ≤ µ′ we have rk (η) ≥ rk (η), and in particular rk (η) ≥
T,µ T,µ′ T
rk (η)
T,µ
2 rk (η)=∪{rk (η⌢hγi)+1:η⌢hγi∈T}.
T T
3 rk (hi)=α.
ds(α)
4 If rk (η)≥α, µ≥α then we can embed η⌢ds(α) into T, so that ρ7→ρ for ρEη.
T,µ
Proof.
3 The proof is by induction on α.
For α=0 this is obvious. Assume correctness for every β <α. ds(α)= {hβi⌢ν :
βS<α
ν ∈ds(β)}.Foreveryβ <α,ν ∈ds(β)wehaverk (hβi⌢ν)=rk (ν),therefore
ds(α) ds(β)
(the last equality is due to the induction hypothesis):
Uniforming n-place functions on T ⊆ds(α) 5
∪{rk (hβi⌢ν)+1:ν ∈ds(β)} = ∪{rk (ν)+1:ν ∈ds(β)}
ds(α) ds(β)
= rk(ds(β))
= β
We therefore have rk(ds(α))=∪{β+1:β <α}=α
4 The proof is by induction on α.
For α=0 there is nothing to prove.
Assume correctness for every β < α, and rk (η) ≥ α, α ≤ µ. For β < α let
T,µ
C ={γ :rk (η⌢hγi)≥β},so|C |≥µandC ⊆C forβ′ <β <α.Byinduction
β T,µ β β β′
on β <α we can choose an increasingsequence of ordinals γ such that γ =minΓ
β β β
where Γ ={γ ∈C :(∀β′ <β)(γ >γ )}. Assume towards contradiction that Γ is
β β β′ β
empty,andletC′ =hγ :β′ <βi∩C .Foreveryγ ∈C \C′ (andthere issuchγ as
β β′ β β β
|C | ≥ µ whereas |C′| ≤ |β| < µ) as γ ∈/ Γ then there is β′ < β such that γ < γ ,
β β β β′
assume β′ is minimal with this property, but that contradicts the choice of γ .
β′
Bytheinductionhypothesisforeveryβ <αthereisϕ whichembeds(η⌢hγ i)⌢ds(β)
β β
in T so that ϕ ↾{ρ : ρ E η⌢hγ i} = Id. We now define ϕ : η⌢ds(α) → T in the
β β α
following manner, if ρEη then ϕ (ρ)=ρ, else ρ=η⌢ν for some ν ∈ds(α), so there
α
is β <α such that ν =hβi⌢ν with ν ∈ds(β), and we define
1 1
ϕ (ρ)=ϕ (η⌢hγ i⌢ν ).
α β β 1
ϕ obviously preserves level.
α
For ρ ⊳ρ in η⌢ds(α) if ρ E η then obviously ϕ (ρ )⊳ϕ (ρ ), and otherwise for
1 2 1 α 1 α 2
some β < α we have ρ = η⌢hβi⌢ν , i ∈ {1,2}, ν ⊳ν ∈ ds(β), and as ϕ is an
i i 1 2 β
embedding we have:
ϕ (ρ )=ϕ (η⌢hγ i⌢ν )⊳ϕ (η⌢hγ i⌢ν )=ϕ (ρ ).
α 1 β β 1 β β 2 α 2
For ρ ∈ η⌢ds(α), γ < γ ordinals such that for i = 1,2 ρ = ρ⌢hγ i ∈ η⌢ds(α),
1 2 i i
necessarily η E ρ and there are β ≤ β < α, ν ∈ ds(β ) so that ρ = η⌢hβ i⌢ν . If
1 2 i i i i i
β =β =β then ν <∗ ν , and as ϕ is an embedding,
1 2 1 ℓx 2 β
ϕ (ρ )=ϕ (η⌢hγ i⌢ν )<∗ ϕ (η⌢hγ i⌢ν )=ϕ (ρ )
α 1 β β 1 ℓx β β 2 α 2
Onthe otherhand,ifβ 6=β thenϕ (ρ )(lg(η))=γ ,andasγ <γ ,alsointhis
1 2 α i βi β1 β2
case ϕ (ρ )<∗ ϕ (ρ ).
α 1 ℓx α 2
By Observation 0.6 ϕ is an embedding, and by definition ϕ ↾{ρ:ρEη}=Id.
α α
The following theorem was was proved By Komja´th and Shelah in [3]:
Theorem 0.10. Assume α is an ordinal and µ a cardinal. Set λ = (|α|µℵ0)+, and
let F : ds(λ+) → µ. Then there is an embedding ϕ : ds(α) → ds(λ+) and a function
c:ω →µ such that for every η ∈ds(α) of length n+1
F(ϕ(η))=c(n).
6 E. Gruenhut and S. Shelah
Inwhatfollowswewillgeneralizetheabovetheorem,intheprocesswewilluseinfinitary
logics. For the readers convenience we include the following definitions.
Definition 0.11.
1 For infinite cardinals κ,λ, and a vocabulary τ consisting of a list of relation and
function symbols and their ‘arity’ which is finite, the infinitary language Lκ,λ for τ is
defined in a similar manner to first order logic. The first subscript, κ, indicates that
formulas have < κ free variables and that we can join together < κ formulas by
or , the second subscript, λ, indicates that we can put <λ quantifiers together inVa
rowW.
2 Given a structure B for τ we say that A is an Lκ,λ-elementary submodel (or sub-
structure), and denote A ≺κ,λ B or A ≺Lκ,λ B, if A is a substructure of B in the
regular manner, and for any Lκ,λ formula ϕ with γ free variables and a¯ ∈ γ|A| we
have
B|=ϕ(a¯)⇔A|=ϕ(a¯).
The Tarski-Vought condition for a substructureA of B to be an elementary submodel
is that for any Lκ,λ-formula ϕ with parameters a¯⊆A we have
B|=∃x¯ϕ(x¯a¯)⇒A|=∃x¯ϕ(x¯a¯).
3 A set X is transitive if for every x∈X we have x⊆X.
4 For every set X there exists a minimal transitive set, which is denoted by TC(X),
such that X ⊆TC(X).
5 For an infinite regular cardinal κ we define
H(κ)={X :|TC(X)|<κ}.
Remark 0.12. In this paper the main use of infinitary logic will be in the following
manner:
1 τ will consist of the two binary relations ∈ and <∗, so |L (τ)|=2κ.
κ+,κ+
2 If κ′ ≤κ,λ′ ≤λ and A≺ B then also A≺ B.
κ,λ κ′,λ′
3 ≺ is a transitive relation.
κ,λ
4 Foraninfinitecardinal µletκ=µ+,λ=2µ,soκis regularandλ<κ =λ.Recallthat
for a structure B and X ⊆ kBk such that |X|+τ ≤ λ ≤ B there is an elementary
Lκ,κ submodel A of B of cardinality λ which includes X.
For further reference on this point see [1].
5 If A ≺κ,κ B and x is definable in B over A (i.e with parameters in A) by an Lκ,κ-
formula, then it is also definable in A by the same formula. In particular if A≺ B
κ,κ
and X ⊆|A|,|X|<κ then X ∈|A|.
Definition0.13. Wesay twofinitesequencehη :ℓ<ni,hν :ℓ<niare similar when:
ℓ ℓ
(a) lg(η )=lg(ν ) for ℓ<n.
ℓ ℓ
(b) lg(η ∩η )=lg(ν ∩ν ) for ℓ,m<n.
ℓ m ℓ m
(c) (η <2 η )≡(ν <2 ν ) for ℓ,m<n (equivalently, we could use <1 ).
ℓ ℓx m ℓ ℓx m ℓx
Uniforming n-place functions on T ⊆ds(α) 7
Observation 0.14.
1 Similarity is an equivalence relation and the number of equivalence classes of finite
sequences is ℵ .
0
2 hη ,...,η ,ν′i, hη ,...,η ,ν′′i are similar if
1 k 1 k
(a) η <2 η <2 ...<2 η
1 ℓx 2 ℓx ℓx k
(b) η <2 ν′
k ℓx
(c) η <2 ν′′
k ℓx
(d) lg(ν′)=lg(ν′′)
(e) lg(ν′∩η )=lg(ν′′∩η )
k k
Proof.
1 Similarity is obviously an equivalence relation.
The equivalence class of a finite sequence of ds(∞) is determined by its length n, the
lengths hn :i< ni of its elements, the lengths hn :i,j <ni of their intersections,
i i,j
and a permutation of n (the order of the elements according to <1 ). Therefore for
ℓx
each n < ω there are ℵ equivalence classes of sequences of length n, and so the
0
number of equivalence classes of finite sequences of ds(∞) is ℵ .
0
2 We need to show that lg(ν′∩η )=lg(ν′′∩η ) for every 0<i<k.
i i
η <2 ν′ and η <2 ν′′. If ν′ ⊳η then we also have lg(ν′′ ∩η ) = lg(ν′ ∩η ) =
k ℓx k ℓx k k k
lg(ν′)=lg(ν′′) so ν′′⊳η , and ν′ =ν′′. In this case obviously the required sequences
k
are similar, so we can assume that there is ℓ such that η ↾ℓ=ν′↾ℓ and ν′(ℓ)>η (ℓ).
k k
By the same reasoning as above we deduce that η ↾ℓ = ν′′↾ℓ and ν′′(ℓ) 6= η (ℓ) so
k k
necessarily ν′′(ℓ)>η (ℓ).
k
The last term we will need before moving on to the main theoremis that of uniformity.
Definition 0.15. Let T ⊆ ds(∞) be a tree, c : [T]<ℵ0 → C. We identify u ∈ [T]<ℵ0
with the <2 -increasing sequence listing it.
ℓx
1 We say T is c-uniform if for any similar u1,u2 in [T]<ℵ0 we have c(u1)=c(u2).
2 We say T is c-end-uniform (or end-uniform for c) when
if η <2 η <2 ... <2 η <2 ρ′,ρ′′ are in T and lg(ρ′) = lg(ρ′′),lg(η ∩ρ′) =
1 ℓx 2 ℓx ℓx k ℓx k
lg(η ∩ρ′′) (equivalently hη ...η ,ρ′i,hη ...η ,ρ′′i are similar-see 0.4(3))
k 1 k 1 k
then c(hη ...η ,ρ′i)=c(hη ,...,η ,ρ′′i).
1 k 1 k
3 We say T is c-n-end-uniform (or n-end-uniform for c) when for k < ω, η ,ρ′,ρ′′ ∈
i j j
ds(∞) (0<i≤k,0<j ≤n) such that
η <2 <η <2 ...<2 η <2 ρ′ <2 ...<2 ρ′
1 ℓx 2 ℓx ℓx k ℓx 1 ℓx ℓx n
η <2 <η <2 ...<2 η <2 ρ′′ <2 <...<ρ′′
1 ℓx 2 ℓx ℓx k ℓx 1 ℓx n
if those two sequences are similar then
c(hη ...,ρ′ ...i)=c(hη ...ρ′′...i).
1 1 1 1
8 E. Gruenhut and S. Shelah
We are now ready for the main theorem of this paper.
Main Claim 0.16. Given a tree S ⊆ ds(∞) and a cardinal µ we can find a tree
T ⊆ds(∞) such that
(∗)1 for every c : [T]<ℵ0 → µ there is T′ ⊆ T isomorphic to S such that c↾T′ is c-end-
uniform.
(∗) |T|<i (|S|+µ).
2 |S|+
Proof. Weassumethat|S|,µareinfinitecardinalssinceoneofourmaingoalsisproving
astatementoftheformx→[y]n ,otherwisetheboundonT hastobeslightlyadjusted.
µ,ℵ0
For each η ∈S let
α =α (η)=otp({ν ∈S :ν <2 η},<2 ),
η S ℓx ℓx
µη =i5αη+1(|S|+µ),
λη =i3(µη)+.
Notethatµ ,λ arethemaximalones,andletχ>>λ ,and<∗ beawellorderingof
hi hi <> χ
H(χ) (see 0.11(5)). By definition, for every η,ν ∈S such that η <2 ν we have µ <µ ,
ℓx η ν
and λ <λ in the following we examine the relation between µ and λ for η 6=ν.
η ν ν η
Observation 0.17. For η <2 ν we have µ ≥λ+.
ℓx ν η
Proof. Since α ≥α +1 we have:
ν µ
µν = i5αν+1(|S|+µ)
≥ i (|S|+µ)
5(αη+1)+1
= i5(µη)
≥ i3(µη)++
= λ+
η
letT :=ds(λ+),wewillshowthatT isasrequired.ObviouslyT meetsrequirement(∗) ,
hi 2
and let c : [T]<ℵ0 → µ. Because of the many details in the following construction we
bring it as a separate lemma.
Lemma 0.18. For η ∈ S we can choose M , T∗ and ν ∈ T for n < ω with the
η η η,n
following properties:
1 Mη is an Lµ+η,µ+η-elementary submodel of B=(H(χ),∈,<∗χ).
2 kMηk=2µη.
3 S,T,c∈M .
η
4 M ,ν ∈M for ρ<∗ η, n<ω.
ρ ρ,n η ℓx
5 Properties of T∗:
η
(a) T∗ =ν ⌢T′ where T′ is isomorphic to ds(22µη).
η η,lg(η)
(b) If ν′,ν′′ ∈ T∗ and are of the same length then they realize the same L -type
η µ+η,µ+η
over M .
η
Uniforming n-place functions on T ⊆ds(α) 9
6 Properties of the ν :
η,n
(a) ν ∈T is of length n.
η,n
(b) ν ∈M .
η,lg(η) η
(c) lg(η)=m<n⇒ν (m)∈/ M .
η,n η
(d) ν ∈T∗, and for n≥lg(η) has at least µ immediate successors in T∗.
η,n η η η
7 If η =η ⌢hαi, then
1
(a) M ,T∗,ν ∈M for n<ω.
η η η,n η1
(b) νη1,n,νη,n realize the same Lµ+η,µ+η-type over {Mρ,νρ,n :n<ω,ρ<∗ℓx η}.
(c) ν =ν for n≤lg (η ).
η1,n η,n 1
(d) ν <∗ ν for n=lg(η).
η,n ℓx η1,n
(e) ν =ν ⌢hγi for some γ.
η,lg(η) η,lgη1
(f) If η′ =η ⌢hα′i with α′ <α then ν <∗ ν .
1 η′,lg(η′) ℓx η,lg(η)
Proof. We show a construction for such a choice by induction on <1 , yes, <1 not
ℓx ℓx
<2 .
ℓx
As the induction is on <1 the base of the induction is the case η = hi. First choose
ℓx
Mhi ≺Lµ+,µ+ B of cardinality 2µhi, so that S,T,c ∈ Mhi (this can be done, see Remark
hi hi
µ+>
0.12). The number of Lµ+,µ+ formulas ϕ(x¯,a¯) where a¯ ⊆ hi Mhi (sequences of length
hi hi
< µ+hi in Mhi) is ≤ (2µhi)µhi = 2µhi hence the number of Lµ+,µ+-types over Mhi is at
hi hi
mostµ′ =22µhi,sowecolourT =ds(λ+)by ≤µ′ colours,c :T →µ′,sothatforρ∈T
hi hi
its colour, c (ρ), codes the L -type which ρ realizes in B over M . As
hi µ+,µ+ hi
hi hi
((i (µ ))µ′ℵ0)+ =i (µ )+ =λ
2 hi 3 hi hi
by Theorem 0.10 there is an embedding of ds(i (µ )) in T, and define T∗ to be its
2 hi hi
image, so that types of sequences from T∗ depend only on their length. We choose
hi
representatives hν : 0 < n < ωi from each level larger than 0 so that for n > 0 ν
hi,n hi,n
and has at least µ immediate successors in T∗ and satisfies 6(c). The latter can be
hi hi
done by cardinality considerations, kMhik = 2µhi, while the cardinality of levels in Tη∗
hi
is i (µ ). We let ν =hi.
2 hi hi,0
It is easily verified that for η =hi all the requirements of the construction are met.
We now show the induction step.
Assume η = η ⌢hα i, lg(η ) = r, and that we have defined for η (and below by <1 )
1 1 1 1 ℓx
and we define for η.
⊛ Let A ={M ,ν :n<ω,ρ<∗ η}.
1 η ρ ρ,n ℓx
For any ρ <∗ η if ρ = η ⌢hαi for some α < α then from requirement (7)(a) of the
ℓx 1 1
construction for ρ we have M ∈ M , and also for all n < ω ν ∈ M , else ρ <∗ η
ρ η1 ρ,n η1 ℓx 1
thereforefromrequirement(4)oftheconstructionforη wehaveforalln<ων ∈M ,
1 ρ,n η1
and Mρ ∈Mη1. So Aη ⊆Mη1, and |Aη|≤µη1, so Aη is definable by an Lµ+η1,µ+η1-formula
with parameters in M , so we have:
η1
10 E. Gruenhut and S. Shelah
⊛ A ⊆M ,|A |≤µ ≤µ , therefore A ∈M .
2 η η1 η η η1 η η1
For every n<ω let
⊛3 ϕn(x)=ϕµη1,n(x)= ( the Lµ+η,µ+η − type which νη1,n realizes over Aη)
And let V
⊛ T ={ρ∈T :B|=ϕ (ρ)}.
4 ϕ lg(ρ)
As the cardinality of the Lµ+η,µ+η-type of any ν ∈B over Aη is at most 2µη which is less
than µη1, for every n < ω we have that ϕn is an Lµ+η1,µ+η1-formula and therefore Tϕ is
definable in Mµη1 by an Lµ+η1,µ+η1-formula, namely
ρ∈T ↔ ρ∈T ∧ (lg(ρ)=n∧ϕ (ρ))
ϕ n
(cid:16) (cid:0)n_<ω (cid:1)(cid:17)
So
⊛ T ∈M and for every n<ω we obviously have ν ∈T .
5 ϕ η1 η1,n ϕ
Recall that for all n < ω ν ∈ T∗, so for any ρ ∈ T∗ of length n, we have that ρ
η1,n η1 η1
realizes the same Lµ+η1,µ+η1-type over Mη1 as νη1,n so in particular they realize the same
Lpaµr+ηt,iµc+ηu-ltayrpϕe o(vxe)r⊢Aηϕ, s(ox↾ρn∈).TIfϕρ. F∈orTm, ≥lgρn=νηm1,ns,oνηB1,m|=↾nϕar(eρo)fththeeresfaomreeBle|n=gtϕh,(sρo↾nin)
m n ϕ m n
and therefore also ρ↾n∈T . We summarize:
ϕ
⊛ T is a subtree of T and T∗ ⊆T .
6 ϕ η1 ϕ
The following point is a crucial one, we show that:
⊛ rk (ν )>µ for every n such that lg(η )≤n<ω .
7 Tϕ,µη1 η1,n η1 1
Assume toward contradiction that rk (ν ) ≤ µ for some lg(η ) ≤ m < ω, and
Tϕ,µη1 η,m η1 1
define for each n such that m≤n<ω :
γ =rk (ν ) and γ∗ =rkµη1 (ν )
n Tϕ,µη1 η,n n Tϕ,µη1 η1,n
(see Definitions 0.7 and 0.8). We now prove by induction on n ≥ m that γ ≤ µ ,
n+1 η1
i.e γ = γ∗. For n = m this is our assumption, and assume that it is known for n. The
n n
following can be expressed by Lµ+η1,µ+η1-formulas with parameters in Mη1 :
ψ : ‘x has rkµη1 (x)=γ ’
1 Tϕ,µη1 n
ψ : ‘x has at least µ immediate successors y in T with rkµη1 (y)≥γ∗ ’
2 η1 ϕ Tϕ,µη1 n+1
We have B |= ψ (ν ), and since T∗ ⊂ T (see ⊛ ) we also have B |= ψ (ν ).
1 η1,n η1 ϕ 6 2 η1,n
By the induction hypothesis for η we have ν ,ν ↾n ∈ T∗ and as they are the
1 η1,n η1,n+1 η1
smaomree ldeentgatihl, rweealihzaevtehtehsaatmrekµLη1µ+η1,µ(+ην1-type↾onv)er=Mγη1,,is.eo rBk|= ψ1(∧νψ2(νη↾1n,n)+1=↾nγ),,oarnind
ν ↾n has at least µ imTmϕ,eµdη1iateη1s,nu+cc1essors innT witThϕ,rµeηd1ucηe1d,nr+a1nk γ∗ ,nso by
η1,n+1 η1 ϕ n+1
the definition of rank (Definition 0.7) we have γ > γ∗ . By the induction hypothesis
n n+1
γ ≤ µ , therefore also γ∗ = γ . In particular we can deduce that γ < γ ,
n η1 n+1 n+1 n+1 n
so having carried out the induction we have an infinite decreasing sequence of ordinals
which is a contradiction.
Recall that lg(η )=r so lg(η)=r+1,
1