Exact saturation in pseudo-elementary classes for simple and stable theories
aa r X i v : . [ m a t h . L O ] S e p EXACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLEAND STABLE THEORIES
ITAY KAPLAN, NICHOLAS RAMSEY, AND SAHARON SHELAH
Abstract.
We study PC-exact saturation for stable and simple theories. Among other results,we show that PC-exact saturation characterizes the stability cardinals of size at least continuumof a countable stable theory and, additionally, that simple unstable theories have PC-exactsaturation at singular cardinals, satisfying mild set-theoretic hypotheses, which had previouslybeen open even for the random graph. We characterize supersimplicity of countable theoriesin terms of having PC-exact saturation at singular cardinals of countable cofinality. We alsoconsider the local analogue of PC-exact saturation, showing that local PC-exact saturation forsingular cardinals of countable cofinality characterizes supershort theories. Introduction
A recurrent theme in model theory is the connection between the combinatorics of definablesets–measured by dividing lines like stability, simplicity, and NSOP –and the ability to constructsaturated models of a given first-order theory. This line of work reaches from the early results ofclassification theory to very current research in the study of, e.g., Keisler’s order and the inter-pretability order. Recently, the notion of exact saturation has proved to be especially meaningfulin relation to combinatorial dividing lines. A model is said be be exactly κ -saturated if it is κ -saturated but not κ + -saturated. The study of this notion was begun in [KSS17] which showedthat, modulo some natural set-theoretic hypotheses, among NIP theories, the existence of exactlysaturated models is characterized by the non-distality of the theory and, additionally, showed thatexactly saturated models of simple theories exist. Later, in [KRS20], exact saturation motivatedthe discovery of a new dividing line, namely the unshreddable theories , containing the simple andNIP theories and giving a general setting in which exactly saturated models may be constructed.In this paper, we study PC-exact saturation, giving many new characterizations of dividinglines. We say that that an L -theory T has PC-exact saturation for κ (where “PC” stands for pseudo-elementary class ) if, for any T ⊇ T with | T | = | T | , there is a model M | = T suchthat the reduct M ↾ L is κ -saturated but not κ + -saturated. The PC version turns out to be Mathematics Subject Classification. much more tightly connected to the complexity of the theory T . For example, consider a twosorted L -structure M = (cid:0) X M , Y M (cid:1) where on one sort X , there is the structure of a countablemodel of Peano arithmetic, and on the second sort Y there is a countably infinite set with nostructure, with no relations between the two sorts. From a model-theoretic point of view, M isas complicated as possible, because it interprets PA, but Th ( M ) has an exactly saturated model:interpreting X as any κ -saturated model of PA and Y as a set with exactly κ many elements yieldsa model of Th ( M ) which is not κ + -saturated. Note, however, that Th ( M ) does not have PC-exact saturation for singular cardinals κ . In an expansion of M to M ′ , in a language containinga function symbol for a bijection f : X → Y , any model of N | = Th ( M ) whose reduct to L is κ -saturated will satisfy (cid:12)(cid:12) X N (cid:12)(cid:12) ≥ κ + , as PA has no exactly κ -saturated model (see [KRS20, Lemma5.3 and the comment after]), hence (cid:12)(cid:12) Y N (cid:12)(cid:12) ≥ κ + and therefore N ↾ L is κ + -saturated. It appearsthat, by looking at PC-exact saturation for singular κ , one obtains a condition that more faithfullytracks the complicated combinatorics of the theory. Indeed, in [MS17, Corollary 9.29] Malliarisand the third-named author observe that the question of whether or not a given theory has exactsaturation for singular cardinals depends essentially only on its class in the interpretability order E ∗ , which is well-known as a measure of model-theoretic complexity.One of the motivations of the work here was a question [MS17, Question 9.31] of whetherthe random graph has PC-exact saturation. For an infinite cardinal κ , one may easily constructan exactly κ -saturated model of the random graph: in a κ + -saturated random graph G , one canchoose an empty (induced) subgraph X ⊆ G with | X | = κ and set G ′ = { v ∈ G | | N G ( v ) ∩ X | < κ } ,where N G ( v ) denotes the neighbors of the vertex v . It is easily checked that, because G was chosento be κ + -saturated, G ′ is a κ -saturated random graph which omits the type { R ( x, v ) | v ∈ X } andhence is not κ + -saturated. Given an expansion of G to a larger language L , it is less clear thatone can arrange for such a G ′ to be a reduct of a model of Th L ( G ) . We prove the existence ofPC-exact saturated models of the random graph by proving a much more general result, showingthat, modulo natural set-theoretic hypotheses, one may construct PC-exact saturated models ofsimple theories and, along the way, we obtain many precise equivalences between subclasses of thestable and simple theories.In Section 3, we begin our study of PC-exact saturation by focusing on the stable theories. Herewe show that, for a countable stable theory T and cardinal µ ≥ ℵ , having PC-exact saturationat µ is equivalent to T being stable in µ ; this is Corollary 3.11. One direction of this theoremis an easy consequence of the well-known fact that stable theories have saturated models in thecardinals in which they are stable and such models may be expanded to a model of any largertheory (of the same size). The other direction involves the construction of suitable expansionsthat allow one to find realizations of types over sets of size κ from realizations of types of smaller XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 3 size. Although it does not appear in the statement, our proof relies on a division into cases basedon whether or not the given stable theory has the finite cover property.In Section 4, we consider simple unstable theories. In general, when T is unstable then it hasPC-exact saturation at any large enough regular cardinal (Proposition 2.6), so we concentrateon singular cardinals here. We show that simple unstable theories have PC-exact saturation forsingular cardinals κ , satisfying certain natural hypotheses. In particular, this implies that therandom graph has singular PC-exact saturation, answering [MS17, Question 9.31]. In fact, weprove that for a theory T ⊇ T such that | T | = | T | , there exists M | = T such that M ↾ L is κ -saturated but not even locally κ + -saturated, that is, there is a partial type consisting of κ many instances of a single formula (or its negation) which is not realized; this is Theorem 4.1. Inthe supersimple case we get a converse for singular cardinals of countable cofinality: an unstable L -theory T is supersimple iff T has PC-exact saturation at singular cardinals with countablecofinality (satisfying mild hypotheses); this is Corollary 4.9. Combined with the results describedin the previous paragraph, we get that a countable theory T is supersimple iff for every κ > ℵ of countable cofinality satisfying mild set-theoretic assumptions, T has PC-exact saturation at κ ;this is Corollary 4.10.Finally, in Section 5, we elaborate on the case of local PC-exact saturation at singular cardinals.Here were are interested in determining, for a given L -theory T and singular cardinal κ of countablecofinality, if, for all T ⊇ T with | T | = | T | , there is M | = T such that M ↾ L is locally κ -saturatedbut not locally κ + -saturated. We essentially characterize when this takes place in terms of thenotion of a supershort theory, a class of theories introduced by Casanovas and Wagner to give alocal analogue of supersimplicity [CW02]. A theory is called supershort if every local type doesnot fork over some finite set. The supershort theories properly contain the supersimple theories.The main result of Section 5 shows that if κ is a singular cardinal of countable cofinality, satisfyingnatural hypotheses, then T has local PC-exact saturation for κ if and only T is supershort, givingthe first “outside” characterization of this class of theories.We conclude in Section 6 with some questions concerning possible extensions of the results ofthis paper. 2. Preliminaries
PC classes and exact saturation.
Here we give the basic definitions and facts about(PC-)exact saturation.
Definition 2.1.
Suppose that T is a complete first order L -theory, and let L ⊇ L . Suppose that T is an L -theory. Let P C ( T , T ) be the class of models M of T which have expansions M tomodels of T . PC stands for “pseudo elementary class”. XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 4
Definition 2.2.
Suppose T is a first order theory and κ is a cardinal.(1) Say that T has exact saturation at κ if T has a κ -saturated model M which is not κ + -saturated.(2) Say that a pseudo-elementary class P = P C ( T , T ) has exact saturation at a cardinal κ ifthere is a κ -saturated model M | = T in P which is not κ + -saturated.(3) Say that T has PC-exact saturation at a cardinal κ if for every T ⊇ T of cardinality | T | , P C ( T , T ) has exact saturation at κ .In general we expect that having exact saturation at κ should not depend on κ (up to some settheoretic assumptions on κ ). For example, in [KSS17] the following facts were established: Fact 2.3.
Suppose T is a first order theory.(1) [KSS17, Theorem 2.4] If T is stable then for all κ > | T | , T has exact saturation at κ .(2) [KSS17, Fact 2.5] If T is unstable then T has exact saturation at all regular κ > | T | .(3) [KSS17, Theorem 3.3] Suppose that T is simple, κ is singular with | T | < µ = cf ( κ ) , κ + = 2 κ and (cid:3) κ holds (see Definition 4.6 below). Then T has exact saturation at κ .(4) [KSS17, Theorem 4.10] Suppose that κ is a singular cardinal such that κ + = 2 κ . An NIPtheory T with | T | < κ is distal (see e.g., [Sim13] ) iff it does not have exact saturation at κ . Local (PC) exact saturation.Definition 2.4.
Work in some complete theory T . Given a set ∆ of formulas and a tuple ofvariables x , let L x, ∆ be the set of formulas of the form ϕ ( x ) where ϕ is any formula in theBoolean algebra generated by ∆ (where by ϕ ( x ) we mean a substitution of the variables in ϕ bysome variables from x ). Similarly, given some set A , L x, ∆ ( A ) is the set of formulas of the form ϕ ( x, a ) where ϕ is any formula from L xy, ∆ and a is some tuple from A (here y is a countablesequence of variables). A ∆ -type in variables x over a set A is a maximal consistent collectionof formulas from L x, ∆ ( A ) . A local type over A is a ∆ -type for some finite set of formulas ∆ .The space of all ∆ -types (in finitely many variables) over A is denoted by S ∆ ( A ) . (Sometimesthis notion is only defined when ∆ is a partitioned set of formulas, ∆ ( x, y ) , but here we allow allpartitions.)We say that a structure M is κ -locally saturated if every local type (in finitely many variables)over a set A ⊆ M of size | A | < κ is realized. Say that M is locally saturated if it is | M | -locallysaturated.We define a local analog to Definition 2.2. Definition 2.5.
Suppose T is a first order theory and κ is a cardinal. XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 5 (1) Say that T has local exact saturation at κ if T has a κ -locally saturated model M whichis not κ + -locally saturated.(2) Say that a PC-class P = P C ( T , T ) has local exact saturation at a cardinal κ if there is a κ -locally saturated model M | = T in P which is not κ + -locally saturated.(3) Say that T has local PC-exact saturation at a cardinal κ if for every T ⊇ T of cardinality | T | , P C ( T , T ) has local exact saturation at κ .2.3. PC-exact saturation for unstable theories in regular cardinals.
For unstable theoriesand regular cardinals, the situation is as in Fact 2.3 (2). Because of the following proposition,when we discuss unstable theories, we will subsequently concentrate on singular cardinals.
Proposition 2.6. If T is not stable then T has PC-exact saturation at any regular κ > | T | .Moreover for any T ⊇ T of size | T | , there is a κ -saturated model of T whose reduct to thelanguage L of T is not κ + -locally saturated.Proof. The proof is almost exactly the same as the proof of [KSS17, Fact 2.5].Let M | = T be of size | T | . For i ≤ κ , define a continuous increasing sequence of models M i where | M i +1 | = 2 | M i | and M i +1 is | M i | + -saturated. Hence M κ is κ -saturated and | M κ | = i κ ( | T | ) .As T is unstable, and M κ is i κ ( | T | ) + -universal | S L ( M κ ↾ L ) | > i κ ( | T | ) (for an explanationsee the proof of [KSS17, Fact 2.5]).However, as the number of L -types over M κ which are invariant (i.e., which do not split)over M i is ≤ | Mi | ≤ i κ ( | T | ) , there is p ( x ) ∈ S L ( M κ ) which splits over every M i . Hence foreach i < κ , there is some L -formula ϕ i ( x, y ) and some a i , b i ∈ M κ such that a i ≡ M i b i and ϕ i ( x, a i ) ∧ ¬ ϕ i ( x, b i ) ∈ p . As κ > | T | , there is a cofinal subset E ⊆ κ such that for i ∈ E , ϕ i = ϕ is constant. Let q ( x ) be { ϕ ( x, a i ) ∧ ¬ ϕ ( x, b i ) | i ∈ E } . Then the local type q is not realized in M κ . (cid:3) PC-Exact saturation for stable theories
The goal of this section is Theorem 3.10: assuming that T is a strictly stable countable completetheory in the language L and µ is a cardinal in which T is not stable, we will find a countable T ⊇ T in the language L ⊇ L such that if M | = T and M ↾ L is ℵ -saturated and locally µ -saturated, then M ↾ L is µ + -saturated.3.1. Description of the expansion.
We will define our desired theory T by choosing a certainmodel of T , describing an expansion, and taking T to be the theory of this structure in theexpanded language. We will assume that L is disjoint from all symbols we are about to present.As T is not superstable, we can use the following fact. XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 6
Fact 3.1. [CR16, Proposition 3.5] If κ ( T ) > ℵ (namely, T is not supersimple), then there is asequence of formulas h ψ n ( x, y n ) | n < ω i (where x is a single variable and the y n ’s are variables ofvarying lengths) and a sequence h a η | η ∈ ω <ω i such that: – a η is an (cid:12)(cid:12) y | η | (cid:12)(cid:12) -tuple; for σ ∈ ω ω , { ϕ n ( x, a σ ↾ n ) | n < ω } is consistent; for every η ∈ ω n , ν ∈ ω m such that η ⊥ ν , { ϕ n ( x, a η ) , ϕ m ( x, a ν ) } is 2-inconsistent. Fix a countable model M | = T such that h a η | η ∈ ω <ω i is contained in M , and we will describeour expansion. Since M is countable we may assume that its universe is ω ∪ ω <ω ∪ P fin ( L ) where P fin ( L ) is the set of all finite subsets of formulas from L in a fixed countable set of variables { v i | i < ω } . First expand M by adding a predicate N for ω and adding + , · , < on N and abijection e : N → M . We also have a predicate T for ω <ω on which we add the order E . Addtwo functions l : T → N and eval :
T × N → N such that l is the length function and eval is the function defined by eval ( η, n ) = η ( n ) for n < l ( η ) and otherwise eval ( η, n ) = 0 (outsideof their domain we define them in an arbitrary way). Note that, in terms of this structure, if η ∈ T and n ∈ N , the concatenation η ⌢ h n i is defined as the unique element ν of T of length l ( η ) + 1 such that eval ( ν, l ( η )) = n and for k < l ( η ) , eval ( ν, k ) = eval ( η, k ) . We similarly define h n i ⌢ η . For notational simplicity, when η ∈ T and k < l ( η ) , we will write η ( k ) instead of eval ( η, k ) . Additionally, we will always refer to N for the natural numbers predicate and use ω for the standard natural numbers. For convenience we add a predicate M for the universe.We will write n n for the definable set of η ∈ T such that l ( η ) = n and for all k < n , η ( k ) < n .Let i be a function with domain { ( η, n ) | η ∈ n n , n ∈ N } and range N such that i ( − , n ) : n n → N is an injection onto an initial segment of N .We will add a predicate L for P fin ( L ) , together with ⊆ giving containment and a truth pred-icate; formally, the set of formulas L is identified with atoms in the Boolean algebra L and thetruth valuation is a function T V from L × T to { , } such that T V ( { ϕ } , η ) = 1 iff ϕ holds in M with the assignment v n e ( η ( n )) for n < l ( η ) and v n otherwise.Add a function d : N → L mapping n to (cid:8) ϕ n (cid:0) v ; v , . . . , v | y n | (cid:1)(cid:9) (where the formulas ϕ n are from above). Let a : T → T be a function such that if η ∈ ω <ω then a ( η ) ∈ ω | y l ( η ) | and { T V ( d ( i ) , h x i ⌢ a ( η | i )) | i < l ( η ) } is consistent and such that if η, ν ∈ T are incomparable then { T V ( d ( l ( η )) , h x i ⌢ a ( η )) , T V ( d ( l ( ν )) , h x i ⌢ a ( ν )) } is inconsistent. This is a direct translationof the properties of the formulas ϕ n described above.We add a bijection c : N → L that associates to each natural number a finite set of formulas.We add a predicate P ⊆ T × L such that ( η, ∆) ∈ P M if and only if h e ( η ( i )) | i < l ( η ) i is a ∆ -indiscernible sequence that extends to a ∆ -indiscernible sequence of countable length. Now wedefine a function F : T ×L×N → M on M so that, if ( η, ∆) ∈ P M , then F M ( η, ∆ , − ) : N → M XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 7 is a ∆ -indiscernible sequence extending h e ( η ( i )) | i < l ( η ) i . Otherwise, F M ( η, ∆ , m ) is definedarbitrarily.We let M = ( M , N , T , L ) with this additional structure and constants for all elements (sothat models are elementary extensions), and we set T = Th ( M ) and L = L ( T ) .3.2. Properties of T .Lemma 3.2. In T , there is a definable function H : N × N → N such that if M | = T and M ↾ L is ℵ -saturated, then for any function f : ω → N M , there is some m f ∈ N M with the propertythat H M ( m f , n ) = f ( n ) for all n ∈ ω .Proof. Define H : N × N → N as follows:(1) If there is k ∈ N and η ∈ T with l ( η ) = n and | = T V ( d ( n + 1) , h g i ⌢ a ( η ⌢ h k i )) , weset H ( g, n ) = k (note that the second parameter of T V gets an element from T so g should be in N for this to be well-defined).(2) Else, we set H ( g, n ) = 0 .Note that if there are η, η ′ ∈ T of length n and k ′ ∈ N such that | = T V ( d ( n + 1) , h g i ⌢ a ( η ⌢ h k i )) ∧ T V ( d ( n + 1) , h g i ⌢ a ( η ′ ⌢ h k ′ i )) then η ⌢ h k i and η ′ ⌢ h k ′ i must be comparable hence thesame (since this is true in M ). It follows that k = k ′ , which shows that H is well-defined. Also, H is clearly definable.Let M | = T be a model such that M ↾ L is ℵ -saturated. Let f : ω → N M be arbitrary andwe will define a path h η i : i < ω i in T M with l ( η i ) = i + 1 inductively, by setting η = h f (0) i and η i +1 = η i ⌢ h f ( i + 1) i (by this we mean that eval (cid:0) η , M (cid:1) = f (0) , etc.; this is possible todo in any model of T ). By the choice of a , we know that { T V ( d ( i + 1) , h x i ⌢ a ( η i )) | i < ω } isa partial type which in turn implies that { ϕ i +1 ( x, e ( a ( η i ) (0)) , . . . , e ( a ( η i ) ( | y i +1 | − | i < ω } is a partial type and, by ℵ -saturation, this is realized by some m ′ f ∈ M . Unraveling definitions,we have H M (cid:16) e − (cid:16) m ′ f (cid:17) , n (cid:17) = f ( n ) for all n ∈ ω , so let m f = e − (cid:16) m ′ f (cid:17) . (cid:3) Lemma 3.3.
Suppose that T has the fcp, M | = T and M ↾ L is locally µ -saturated. Then forany non-standard n ∈ N M , we have | [0 , n ) | ≥ µ .Proof. As T has the finite cover property, there is a formula ϕ ( x, y ) with x a singleton such thatin any model of T , for all k < ω , there are some k ′ > k and h a i | i ≤ k ′ i such that { ϕ ( x, a i ) | i < k ′ } is inconsistent yet k -consistent. For simplicity assume that y is a singleton (otherwise replace η by a tuple of η ’s in what follows). In particular, for all k < ω , M satisfies χ ( k ) which assertsthat there exists k ′ > k such that k ′ < n and η ∈ T with l ( η ) = k ′ such that the following aresatisfied: XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 8 – ¬∃ x ∈ M (cid:0)V i Suppose M | = T is given such that M ↾ L is ℵ -saturated.(1) If | M | ≥ µ , then | M | ≥ µ ℵ .(2) If, moreover, M has the property that [0 , n ) ≥ µ for all nonstandard n ∈ N M , then forany nonstandard n ∈ N M , [0 , n ) ≥ µ ℵ .Proof. (1) Given any f : ω → N M , as M ↾ L is ℵ -saturated, we know, by Lemma 3.2, there issome m f ∈ N M such that H ( m f , i ) = f ( i ) , for all i < ω . As e M : N M → M M is a bijection, (cid:12)(cid:12) M M (cid:12)(cid:12) = (cid:12)(cid:12) N M (cid:12)(cid:12) so, as there are (cid:12)(cid:12) N M (cid:12)(cid:12) ℵ ≥ µ ℵ functions f : ω → N M and the map f m f is clearly injective, we have | M | ≥ µ ℵ , which proves(1).Now we prove (2). Recall that in M | = T , we defined the function i such that, for all n , i ( − , n ) : n n → N is an injection onto an initial segment of N .Let log : N → N be the (definable) function given by log ( n ) = max { k | ran ( i ( − , k )) ⊆ [0 , n ) } . On the standard natural numbers, log ( n ) is the largest k such that k k ≤ n .Given n nonstandard, let k = log M ( n ) . Because i ( − , k ) gives an injection of (cid:0) k k (cid:1) M into [0 , n ) ,we are reduced to showing that (cid:12)(cid:12)(cid:12)(cid:0) k k (cid:1) M (cid:12)(cid:12)(cid:12) ≥ µ ℵ . Note that k is also nonstandard, so by hypothesis, | [0 , k ) | ≥ µ . Let f : ω → [0 , k ) be an arbitrary function. As in (1), by Lemma 3.2, there is some m f ∈ N M such that H ( m f , i ) = f ( i ) for all i < ω . XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 9 Note that if j < ω , and m ∈ N M , then there is an element η ∈ j j such that min (cid:8) H M ( m, i ) , j − (cid:9) = η ( i ) for all i < j , and hence M | = ∀ j ∈ N ∀ m ∈ N ∃ η ∈ j j [( ∀ i < j ) [ η ( i ) = min { H ( m, i ) , j − } ]] , and hence this sentence is contained in T . Applying this with j = k and m = m f , we know thereis ˜ η f ∈ (cid:0) k k (cid:1) M such that ˜ η f ( i ) = H ( m f , i ) , for all i < ω (using that H ( m f , i ) = f ( i ) < k for all i < ω ). Moreover, because ˜ η f ( i ) = f ( i ) forall standard i , we have f = f ′ implies ˜ η f = ˜ η f ′ . This shows (cid:12)(cid:12)(cid:12)(cid:0) k k (cid:1) M (cid:12)(cid:12)(cid:12) ≥ µ ℵ , which completes theproof. (cid:3) Fact 3.5. [She90, Theorem II.4.6] Suppose T is nfcp, ∆ is a finite set of formulas. There is an n (∆) < ω such that, if A is a set of parameters and { a γ | γ < α } is a ∆ -indiscernible set over A and n (∆) ≤ α < β , then there exist a γ for α ≤ γ < β such that { a γ | γ < β } is a ∆ -indiscernibleset over A .Remark . In [She90, Theorem II.4.6], the result is a bit more refined since it applies to ∆ - n -indiscernible sequences and gives one n (∆) which works for all n (note that being ∆ -indiscernibleis equivalent to being ∆ - n -indiscernible for some large enough n ), but this is not needed here. Remark . If, in the context of the previous fact, M is a model and we are given that A is finiteand α < β ≤ ω and Aa <α ⊆ M , then we can choose a γ ∈ M for α ≤ γ < β (if β = ω we extendone by one). Likewise, if Aa <α ⊆ M and M is ( | A | + | β | ) + -saturated, we may choose a γ ∈ M for α ≤ γ < β . Definition 3.8. The infinite indiscernible sequences I and I are equivalent if there is an infiniteindiscernible sequence J such that I ⌢ J and I ⌢ J are both indiscernible. If M is a modeland I is an infinite indiscernible sequence contained in M , we define dim ( I, M ) by dim ( I, M ) = min {| J | | J maximal indiscernible in M equivalent to I } . If dim ( I, M ) = | J | for all maximal indiscernible sequences in M equivalent to I , then we say thatthe dimension is true . Fact 3.9. Suppose T is a countable stable theory and M | = T .(1) [She90, Theorem III.3.9] For any infinite indiscernible I contained in M , the dimension dim ( I, M ) is true (there it is stated with the added condition that dim ( I, M ) ≥ κ ( T ) but this condition is not necessary when T is countable as follows from the proof there;however since we will later assume ℵ -saturation we can just apply the reference as is). XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 10 (2) [She90, Theorem III.3.10] Assume that M is ℵ -saturated. For an infinite cardinal λ , M is λ -saturated if and only if for every infinite indiscernible sequence I in M of singleelements, we have dim ( I, M ) ≥ λ . (In the reference there is no restriction on the lengthof the tuples, but it is enough to consider sequences of elements as can be seen from theproof there and the fact that saturation is implied by realizing 1-types.) Theorem 3.10. Suppose T is a strictly stable countable theory and µ is a cardinal such that T isnot stable in µ . Then there is a countable theory T ⊇ T with the property that for all M | = T , if M ↾ L is ℵ -saturated and locally µ -saturated, then M ↾ L is µ + -saturated.Proof. As T is stable in every cardinal κ satisfying κ | T | = κ , we have µ ℵ > µ . Let T be thetheory described above. Let M | = T be arbitrary with the property that M ↾ L is ℵ -saturatedand locally µ -saturated. By Fact 3.9(1), the dimension of any infinite indiscernible sequence in M is true. Since every infinite indiscernible sequence is equivalent to all of its infinite initial segments,we know that, by Fact 3.9(2), to show M ↾ L is µ + -saturated, it suffices to show that every length ω indiscernible sequence (with respect to the language L ) contained in M extends to one that haslength ≥ µ + . Note that M ≻ M so in particular contains all standard numbers and finite sets offormulas.Fix an arbitrary indiscernible sequence I = h a i | i < ω i of elements in M and we will findan extension of length ≥ µ + . For each i < ω , there is some η i ∈ T M with l ( η i ) = i and h e ( η i ( j )) | j < i i = h a j | j < i i . We fix also an increasing sequence of finite sets of formulas ∆ n such that L = S n<ω ∆ n .Case 1. For all n < ω , M | = P ( η n , ∆ n ) .As M ↾ L is, in particular ℵ -saturated, we may apply Lemma 3.2 to find m , m ∈ N M suchthat H ( m , n ) = e − ( a n ) and H ( m , n ) = c − (∆ n ) for all n < ω . For n ∈ N M , let ν ( n ) bethe element of T such that l ( ν ( n )) = n and eval ( ν ( n ) , i ) = H ( m , i ) for all i < n . Notice thatwe have that the function n ν ( n ) is definable in M and ν ( n ) = η n for all (standard) n < ω .Next, let ∆ ( n ) = c ( H ( m , n )) for all n ∈ N M . Likewise, we have that the function n ∆ ( n ) is M -definable and ∆ ( n ) = ∆ n for all n < ω .By our assumptions, we have that for all n < ω , M | = ∀ k ≤ n (∆ ( k ) ⊆ ∆ ( n ) ∧ ν ( k ) E ν ( n )) ∧ P ( ν ( n ) , ∆ ( n )) , and, hence, by overspill (which we may apply since nonstandard elements exists by ℵ -saturation),there is some nonstandard n ∗ ∈ N M such that M | = ∀ k ≤ n ∗ (∆ ( k ) ⊆ ∆ ( n ∗ ) ∧ ν ( k ) E ν ( n ∗ )) ∧ P ( ν ( n ∗ ) , ∆ ( n ∗ )) . If the reader is not satisfied with this, they can alter the language L by adding predicates P n and functions F n to deal with sequences of length n , for all n < ω . XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 11 Since n ∗ is nonstandard, we know that ∆ ( n ∗ ) contains all standard formulas of L and ν ( n ∗ ) ex-tends ν ( k ) for all k < ω . Since, additionally, M | = P ( ν ( n ∗ ) , ∆ ( n ∗ )) , it follows that F M ( ν ( n ∗ ) , ∆ ( n ∗ ) , − ) : N M → M enumerates an L -indiscernible sequence extending I . (Note that this was a property of F M and using the truth predicate the fact that this is an indiscernible sequence is expressible in L so this is also true in M ; this also uses the fact that for standard formulas, the truth predicategives the correct answer.)The local µ -saturation of M ↾ L implies | M | ≥ µ , so, by Lemma 3.4(1), we have (cid:12)(cid:12) N M (cid:12)(cid:12) = | M | ≥ µ ℵ ≥ µ + , we have shown that I extends to an indiscernible sequence of length µ + .Case 2. There is an N < ω such that M | = ¬ P ( η N , ∆ N ) .Note that it follows that for all ω > r ≥ N , M | = ¬ P ( η r , ∆ N ) (since η r extends η N and thisimplication is true in M ). If T was nfcp, then in particular this is true for any r ≥ N, n (∆ N ) ,where n (∆ N ) is from Fact 3.5. Thus by elementarity, M | = ∃ η ∈ T [ l ( η ) = r ∧ ¬ P ( η, ∆ N ) ∧ h e ( η ( i )) | i < r i is ∆ N -indiscernible ] hence T has the finite cover property by choice of r and Remark 3.7. However, because I is L -indiscernible, we have ν ( n ) is ∆ n -indiscernible for all n < ω . Therefore, we have that for all n < ω , M satisfies the conjunction of the following sentences (which can be expressed using thetruth predicate): ∀ k ≤ n ((∆ ( k )) ⊆ ∆ ( n ) ∧ ν ( k ) E ν ( n )) . ∀ k ≤ n [ h e ( ν ( n ) ( i )) | i < n i is ∆ ( n ) -indiscernible ] .Thus, by overspill, there is some nonstandard n ∗ ∈ N M such that h e ( ν ( n ∗ ) ( i )) | i < n ∗ i is L -indiscernible and extends I . By Lemma 3.3 and Lemma 3.4(2), we know | [0 , n ∗ ) | ≥ µ ℵ > µ , sowe have shown I extends to an indiscernible sequence of length ≥ µ + .This completes the proof. (cid:3) Corollary 3.11. For a countable stable theory T and µ ≥ ℵ , the following are equivalent:(1) T has PC-exact saturation in µ .(2) T is stable in µ .Proof. (1) implies (2): if T is superstable then (2) holds automatically. Otherwise, we are doneby Theorem 3.10.(2) implies (1): if T is stable in µ , then by [She90, Chapter VIII, Theorem 4.7] T has a saturatedmodel M of size µ . Since saturated models are resplendent [Poi00, Theorem 9.17] we can expand M to a model M ′ of T . (cid:3) XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 12 PC-exact saturation for simple and supersimple theories Simple theories. This section is devoted to the proof of the following theorem, which inparticular answers [MS17, Question 9.31] (about the random graph). Theorem 4.1. Assume that T is a complete simple unstable L -theory, T ⊇ T is a theory in L ⊇ L , | T | ≤ | T | . Also, assume that κ is a singular cardinal such that κ ( T ) ≤ µ = cf ( κ ) , | T | < κ , κ + = 2 κ and (cid:3) κ holds (see Definition 4.6). Then P C ( T , T ) has exact saturation at κ .Moreover, there is a model M | = T whose reduct to L is κ -saturated but not κ + -locally saturated. The proof is somewhat similar to the proof of the parallel theorem from [KSS17] (Fact 2.3 (3)),with some important differences. The class M from there is similar to the class C here, and theoverall structure of the proof is similar, but the proof of the main lemma (Lemma 4.3 below) isquite different.We may assume that T has built-in Skolem functions. Since T is unstable and simple it has theindependence property, as witnessed by some formula ϕ ( x, y ) from L . Suppose that κ = P i<µ λ i where the sequence h λ i | i < µ i is continuous, increasing, and each λ i is regular for i < µ a successor.Also, assume that | T | ≤ λ (here we use the assumption that | T | < κ ). We work in a monstermodel C of T , and denote its L -reduct by C . Let I = h a α | α < κ i be an L -indiscernible sequenceof y -tuples of order type κ , which witnesses that ϕ has IP. For i < µ , let I i = h a α | α < λ i i . Also,for α < κ , let ¯ a α be the sequence h a ωα + k | k < ω i . Definition 4.2. Let C be the class of sequences h A i | i < µ i such that:(1) For all i < µ , A i ≺ C ; | A i | ≤ λ i ; I i ⊆ A i .(2) The sequence h A i | i < µ i is increasing continuous.(3) For all i < µ and every finite tuple c ∈ A i +1 , there is a club E ⊆ λ i +1 such that for all α ∈ E , ¯ a α is L -indiscernible over c .For ¯ A, ¯ B ∈ C , write ¯ A ≤ ¯ B for: for every i < µ , A i ⊆ B i .For example, letting A i = Sk ( I i ) (the Skolem hulls of I i ) for i < µ , ¯ A = h A i | i < µ i ∈ C . Then ¯ A is ≤ -minimal. Main Lemma 4.3. Suppose that ¯ A ∈ C and let A = S { A i | i < µ } . Suppose that C ⊆ A has size < κ and p ( x ) ∈ S L ( C ) . Then there is ¯ B ∈ C such that ¯ A ≤ ¯ B and B = S { B i | i < µ } realizes p .Proof. We want to make the following assumptions first. By maybe increasing C by a set of size ≤ | T | + | C | + µ we may assume that: ( ⋆ ) C | ⌣ A i ∩ C A i and A i ∩ C ≺ C for all i < µ . ( | ⌣ denotes non-forking independence.)(This is straightforward, but for a proof see the beginning of the proof in [KSS17, Main Lemma3.11].) XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 13 Without loss of generality assume that p does not fork over A ∗ = C ∩ A (this uses theassumption that cf ( κ ) = µ ≥ κ ( T ) ). (Let i ∗ < µ be minimal successor or 0 such that p does notfork over C ∩ A i , and let λ ′ i = λ i ∗ + i , for i < µ . If the lemma is true for λ ′ i (and A ′ i = A i ∗ + i )instead of λ i , and ¯ B ′ = h B ′ i | i < µ i witnesses this (so | B ′ i | ≤ λ ′ i , etc.), then let B i = A i for i < i ∗ and for i ≥ i ∗ , let j < µ be such that i ∗ + j = i and B i = B ′ j .)Fix an enumeration of A as h b α | α < κ i such that, for i < µ , A i is enumerated by h b α | α < λ i i .For each i < µ , let E i ⊆ λ i +1 be a club such that: – If α ∈ E i then α = ωα ; ¯ a α = h a ωα + k | k < ω i is indiscernible over A <α = { b β | β < α } (which equals A <ωα ); A <α ≺ C ; if α ∈ E i then ¯ a β ⊆ A <α for all β < α and A <α ⊇ C ∩ A i .Its existence is proved as follows. First note that the set of α < λ i +1 for which α = ωα forms a club.Next, since the club filter on λ i +1 is λ i +1 -complete, for every set D ⊆ A i +1 of size < λ i +1 , thereis a club E D ⊆ λ i +1 such that if α ∈ E D then ¯ a α is indiscernible over D . Now let E ′ i be the diag-onal intersection △ β<λ i +1 E A <β = n α < λ i +1 (cid:12)(cid:12)(cid:12) α ∈ T β<α E A <β o and let E i = { α ∈ E ′ i | ωα = α } .Now, if α ∈ E i , then ¯ a α is indiscernible over A <β for all β < α , and since α is a limit (as ωα = α ), ¯ a α is indiscernible over S { A <β | β < α } = A <α . This takes care of the requirementthat ¯ a α is indiscernible over A <α . The other requirements follow since { α < λ i +1 | A <α ≺ C } , { α < λ i +1 | ∀ β < α (¯ a β ⊆ A <α ) } and { α < λ i +1 | C ∩ A i ⊆ A <α } are clubs.Let E = S { E i | i < µ } . Let Γ ( x ) be the set of formulas saying that p ( x ) holds and that for all α ∈ E , ¯ a α is L -indiscernible over A <α ∪ x . Claim . It is enough to show that Γ ( x ) is consistent. Proof. Let d | = Γ ( x ) . Let B ′ i = A i ∪ d for all i < µ . Note that for each i < µ there is a club E i ⊆ λ i +1 such that if α ∈ E i then ¯ a α is indiscernible over A <α d . Now let c ⊆ B ′ i be anyfinite set. Then c ⊆ A <α d for some α < λ i +1 , so E ′ = E i ∩ [ α + 1 , λ i ) is such that for any α ∈ E ′ , ¯ a α is indiscernible over c . Finally, let B i be Sk ( B ′ i ) . Since the indiscernibility was in L , ¯ B = h B i | i < µ i ∈ C . (cid:3) So fix finite sets F ⊆ E , v ⊆ κ , and a finite set of L -formulas ∆ , and let Γ F ,v, ∆ ( x ) say that p ( x ) holds and for all α ∈ F , ¯ a α is ∆ -indiscernible over { b β | β ∈ v ∩ α } ∪ { x } , and we want toshow that Γ F ,v, ∆ is consistent.Let n = | F | , and write F = { α i | i < n } where α < . . . < α n − . Let T be the tree of allfunctions (cid:16) [ ω ] ℵ (cid:17) ≤ n ( [ ω ] ℵ is the set of countably infinite subsets of ω ). For every i < n and everyinfinite s ⊆ ω , let f i,s be a partial elementary map taking ¯ a α i to ¯ a α i ↾ s fixing A <α i (i.e., mapping a ωα i + k to a ωα i + k ′ where k ′ is the k -th element in s ). Note that A ∗ = A ∩ C is fixed by all the f i,s since A ∗ ⊆ A <α for every α ∈ E (by the last requirement on E i in the bullet above). Claim . To show that Γ F ,v, ∆ is consistent, it is enough to prove the following: XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 14 ( † ) There is an assignment, assigning each η ∈ T of height i +1 ≤ n an automorphism σ η of C extending f i,η ( i ) in such a way that, letting τ η = σ η ↾ ◦· · ·◦ σ η ↾ i +1 , Θ ( x ) = S { τ η ( p ) | η ∈ T } is consistent. Proof. Suppose d ∗ | = Θ ( x ) . By Ramsey, there is some infinite s ⊆ ω such that f ,s (¯ a α ) =¯ a α ↾ s is ∆ -indiscernible over { b β | β ∈ v ∩ α } ∪ { d ∗ } . Let η (0) = s . Suppose we chose η ↾ i for some ≤ i < n . Again by Ramsey there is some infinite s i ⊆ ω such that, letting η ( i ) = s i wehave that σ η ↾ ◦ · · · ◦ σ η ↾ i +1 (¯ a α i ) = σ η ↾ ◦ · · · ◦ σ η ↾ i (¯ a α i ↾ s i ) = [ σ η ↾ ◦ · · · ◦ σ η ↾ i (¯ a α i )] ↾ s i is ∆ -indiscernible over σ η ↾ ◦ · · · ◦ σ η ↾ i ( { b β | β ∈ v ∩ α i } ) ∪ { d ∗ } . This procedure defines η ∈ T of height n . Then τ − η ( d ∗ ) | = p and ¯ a α is ∆ -indiscernible over (cid:8) τ − η ( d ∗ ) (cid:9) ∪{ b β | β ∈ v ∩ α } for each α ∈ F as we wanted. Indeed, τ − η ( d ∗ ) | = p obviously. Also, for each i < n , τ − η ( σ η ↾ ◦ · · · ◦ σ η ↾ i +1 (¯ a α i )) = σ − η ↾ n ◦ · · · ◦ σ − η ↾ ( σ η ↾ ◦ · · · ◦ σ η ↾ i +1 (¯ a α i )) = ¯ a α i because for j > i + 1 , σ η ↾ j fixes A <α i +1 which con-tains ¯ a α i . Hence ¯ a α i is indiscernible over the union of σ − η ( d ∗ ) and τ − η ◦ σ η ↾ ◦· · ·◦ σ η ↾ i { b β | β ∈ v ∩ α i } which is just { b β | β ∈ v ∩ α i } because σ η ↾ j fixes A <α i +1 for j > i . (cid:3) Fix some enumeration of T , h η ε | ε < ε ∗ i such that η = ∅ , ε ∗ is a limit and if η ε E η ζ then ε ≤ ζ .Let σ = τ = id and for < ε < ε ∗ define σ ε = σ η ε as in ( † ) and consequently τ ε = τ η ε = σ η ε ↾ ◦ · · · ◦ σ η ε ↾ | η ε | by induction on ε , in such a way that: ( † ǫ ) Θ ε ( x ) = S { τ ζ ( p ) | ζ < ε } is consistent and does not fork over A ∗ .This will obviously suffice in order to prove ( † ) , and holds trivially for ε = 0 by choice of A ∗ .Suppose we have already defined σ ζ for all ≤ ζ < ε , and let η = η ε . By assumption on theorder, we already defined σ η ′ for the predecessor η ′ of η (recall that η = ∅ because < ε ). Assume i + 1 = | η | ≤ n . First, let σ be any automorphism extending f i,η ( i ) and let q be a global coheirextending tp ( τ η ′ ( σ ( C )) /N ) where N = τ η ′ ◦ σ (cid:0) Sk (cid:0) A < αi ¯ a α i (cid:1)(cid:1) . Now let C ′ | = q | Nτ <ε ( C ) where τ <ε ( C ) = S { τ ζ ( C ) | ζ < ε } . Note that σ ( C ) σ (Sk ( A <α i ¯ a α i )) ≡ τ η ′ ( σ ( C )) N ≡ C ′ N ≡ τ − η ′ ( C ′ ) σ (Sk ( A <α i a α i )) . Let σ ′ be an automorphism mapping σ ( C ) to τ − η ′ ( C ′ ) fixing σ (cid:0) Sk (cid:0) A < αi ¯ a α i (cid:1)(cid:1) . Now let σ η = σ ′ ◦ σ . By construction, σ η extends f i,η ( i ) . Note that C ′ = τ η ( C ) | ⌣ uN τ <ε ( C ) (where | ⌣ u means co-heir independence) and that N = τ η (cid:0) Sk (cid:0) A < αi ¯ a α i (cid:1)(cid:1) (because σ ′ ◦ σ (cid:0) Sk (cid:0) A < αi ¯ a α i (cid:1)(cid:1) = σ (cid:0) Sk (cid:0) A < αi ¯ a α i (cid:1)(cid:1) ).Now we check that ( † ) ε +1 holds.By ( ⋆ ) above, we know that C | ⌣ A <αi Sk ( A <α i ¯ a α i ) since A <α i ¯ a α i is contained in the appropri-ate A i (i.e., for an i < µ for which α i ∈ E i ) which is a model, and A <α i contains C ∩ A i . Hence,applying τ η , we get τ η ( C ) | ⌣ τ η ( A <αi ) N . By transitivity, we get that τ η ( C ) | ⌣ τ η ( A <αi ) τ <ε ( C ) .Note that σ η fixes A <α i , so τ η ( A <α i ) = τ η ′ ( A <α i ) . XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 15 By induction, there is some d | = Θ ε ( x ) such that d | ⌣ A ∗ τ <ε ( C ) . Let d ≡ A ∗ τ <ε ( C ) d besuch that d | ⌣ A ∗ τ η ′ ( A <α i ) τ <ε ( C ) . Let d = τ η (cid:16) τ − η ′ ( d ) (cid:17) . Since η ′ comes before η in theenumeration, we have that d | = τ η (cid:16) τ − η ′ ( τ η ′ ( p )) (cid:17) = τ η ( p ) and d | ⌣ A ∗ τ η ( A <α i ) τ η ( C ) (recallthat A ∗ is fixed by σ ζ for all ζ ≤ ε ). Recalling that τ η ( A <α i ) = τ η ′ ( A <α i ) , by base mono-tonicity we have that d | ⌣ τ η ′ ( A <αi ) τ <ε ( C ) , d | ⌣ τ η ′ ( A <αi ) τ η ( C ) , τ η ( C ) | ⌣ τ η ′ ( A <αi ) τ <ε ( C ) and d ≡ τ η ′ ( A <αi ) d (the last equivalence is because τ η ◦ τ − η ′ fixes τ η ′ ( A <α i ) , because τ η and τ η ′ agreeon A <α i , since σ η ↾ A <α i = id ). By the independence theorem for simple theories (see [TZ12,Theorem 7.3.11]), we can find some d ≡ τ η ′ ( A <αi ) τ η ( C ) d (so d | = τ η ( p ) ), d ≡ τ η ′ ( A <αi ) τ <ε ( C ) d (so d | = Θ ε ( x ) ) and d | ⌣ τ η ′ ( A <αi ) τ ≤ ε ( C ) . Finally, since d | ⌣ A ∗ τ η ′ ( A <α i ) , by transitivity wehave that d | ⌣ A ∗ S τ ≤ ε ( C ) . This finishes the proof of the lemma. (cid:3) Now the proof of Theorem 4.1 continues precisely as the proof of [KSS17, Theorem 3.3] withsmall changes.First we recall the definition of (cid:3) κ : Definition 4.6. ( Jensen’s Square principle , [Jec03, Page 443]) Let κ be an uncountable cardinal; (cid:3) κ (square- κ ) is the following condition:There exists a sequence h C α | α ∈ Lim ( κ + ) i such that:(1) C α is a closed unbounded subset of α .(2) If β ∈ Lim ( C α ) then C β = C α ∩ β (where for a set of ordinals X , Lim ( X ) is the set oflimit ordinals in X ).(3) If cf ( α ) < κ , then | C α | < κ . Remark . Suppose that h C α | α ∈ Lim ( κ + ) i witnesses (cid:3) κ . Let C ′ α = Lim ( C α ) . Then thefollowing holds for α ∈ Lim ( κ + ) .(1) If C ′ α = ∅ , then either sup ( C ′ α ) = α , or C ′ α has a last element < α in which case cf ( α ) = ω .If C ′ α = ∅ then cf ( α ) = ω as well.(2) C ′ α ⊆ Lim ( α ) and for all β ∈ C ′ α , C ′ α ∩ β = C ′ β .(3) If cf ( α ) < κ , then | C ′ α | < κ .Define ¯ A ≤ i ¯ B for ¯ A, ¯ B ∈ C and i < µ as A j ⊆ B j for all i ≤ j (so ≤ = ≤ on C ). Write ¯ A ≤ ∗ ¯ B for: there is some i < µ , such that ¯ A ≤ i ¯ B . Proof of Theorem 4.1. Let h C α | α ∈ Lim ( κ + ) i be a sequence as in Remark 4.7. Note that | C α | < κ for all α < κ + as κ is singular. Let { S α | α < κ + } be a partition of κ + to sets of size κ + . Weconstruct a sequence (cid:10)(cid:0) ¯ A α , ¯ p α (cid:1) (cid:12)(cid:12) α < κ + (cid:11) such that:(1) ¯ A α = h A α,i | i < µ i ∈ C (recall that µ = cf ( κ ) );(2) ¯ p α is an enumeration h p α,β | β ∈ S α \ α i of all complete types over subsets of S i A α,i of size < κ (this uses κ + = 2 κ and | T | ≤ κ ); XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 16 (3) If β < α then ¯ A β ≤ ∗ ¯ A α ;(4) If α ∈ S γ and γ ≤ α , then ¯ A α +1 contains a realization of p γ,α ;(5) If α is a limit ordinal, then for all i < µ such that | C α | < λ i and for all β ∈ C α , ¯ A β ≤ i ¯ A α .The construction is done almost precisely as in [KSS17, Proof of Theorem 3.3], but we explain.Put A ,i = Sk ( I i ) . For α successor use Main Lemma 4.3. For α limit, we divide into two cases. Case sup ( C α ) = α . Let i = min { i < µ | | C α | < λ i } (which is a successor). For i < i , let A α,i = A ,i . For i ≥ i successor, let A α,i = S β ∈ C α A β,i . Note that | A α,i | ≤ λ i for all i < µ . We have to show that ¯ A α satisfies (1), (3) and (5). The latter is by construction.For (1), suppose s ⊆ A α,i is a finite set where i ≤ i ∈ Succ ( κ ) . For every element e ∈ s , there is some β e ∈ C α such that e ∈ A β e ,i . Let β = max { β e | e ∈ s } . Then β isa limit ordinal and C α ∩ β = C β . As | C β | < λ i , it follows by the induction hypothesisthat s ⊆ A β,i . Hence for some club E of λ i +1 , ¯ a α is L -indiscernible over s for all α ∈ E . We also have to check that A α,i ≺ C , but this is immediate as A β,i ≺ C forall β ∈ C α by induction.Lastly, (3) is easy by assumption of the case and transitivity of ≤ ∗ . Case sup ( C α ) < α . If C α = ∅ , let γ = 0 and otherwise let γ = max C α (recall that it exists).Let h β n | n < ω i be a cofinal increasing sequence in α starting with β = γ (which existssince cf ( α ) = ω ). For every n < ω , there is some i n < µ such that ¯ A β n ≤ i n ¯ A β n +1 .Without loss of generality assume that i n < i n +1 for all n < ω . Letting i − = 0 ,for all successor i ≥ i n − such that i < i n define A α,i = A β n ,i , and for all successor i ≥ sup { i n | n < ω } , let A α,i = S { A β n ,i | n < ω } . Note that ¯ A β n ≤ i n − ¯ A α for all n < ω . This easily satisfies all the requirements. For example (5): if C α = ∅ , then thereis nothing to check, so assume C α = ∅ . Let i < µ be such that | C α | < λ i and fix some β ∈ C α . Hence β ≤ γ = max C α . Note that ¯ A γ ≤ ¯ A α (so also ¯ A γ ≤ i ¯ A α ), so we mayassume β < γ . In this case, since C γ = C α ∩ γ , β ∈ C γ , and since | C γ | = | C α ∩ γ | < λ i ,by induction it follows that ¯ A β ≤ i ¯ A γ ≤ ¯ A α so we are done.Finally, let M = S α<κ + ,i<µ A α,i . Then M is a κ -saturated model of T by (4). However, it isnot κ + -locally saturated because the local type { ϕ ( x, a j ) | j ∈ I even } ∪ {¬ ϕ ( x, a j ) | j ∈ I odd } isnot realized in M . To see this, suppose towards contradiction that b realizes it. Since ¯ A α is anincreasing continuous sequence for all α < µ + , there must be some α < µ + and i ∈ Succ ( κ ) suchthat b ∈ A α,i . But then by (1), for some α < λ i , ¯ a α is indiscernible over b — contradiction. (cid:3) Supersimple theories. From the previous section it follows that if T is countable, unstableand supersimple ( κ ( T ) = ℵ ), and κ is singular with cofinality ω such that κ + = 2 κ and (cid:3) κ holds, then T has PC-exact saturation at κ . In this section we will show that this propertyidentifies supersimplicity among unstable theories: assuming that T is not supersimple, we will XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 17 find an expansion T (of the same size) such that if M | = T and M ↾ L is κ -saturated, then M is κ + -saturated. Theorem 4.8. Suppose that T is unstable and not supersimple, and that κ is singular with cf ( κ ) = ω (in particular uncountable) and κ ≥ | T | . Then there is an expansion T ⊇ T with | T | = | T | and such that if M | = T is such that M ↾ L is κ -saturated, then M ↾ L is κ + -saturated.Proof. If T is countable we can use essentially the same expansion T as in Section 3.1 (adding toit the function k from below), but since we do not assume that T is countable we give details. Let λ = | T | + ℵ . As T is not supersimple, we can use Fact 3.1 and compactness to find a sequence offormulas h ψ n ( x, y n ) | n < ω i and a tree h a η | η ∈ λ <ω i as there. In addition, since T is not stable,there is a formula ϕ ( x, y ) where x is a singleton and a sequence h b n , c n | n < ω i such that ϕ ( b n , c m ) holds iff n < m .Let M | = T be of size λ containing h a η | η ∈ λ <ω i and h b n | n < ω i . Without loss of generalitywe may assume that the universe of M is λ ∪ λ <ω ∪ L where L is the set of all formulas from L in a fixed countable set of variables { v i | i < ω } . We put a predicate K for λ and a predicate T for λ <ω on which we add the order E . We put a predicate N on ω ⊆ λ and enrich it with + , · , < . Add two functions l : T → N and eval : T × N → K as in Section 3.1: eval ( η, n ) = η ( n ) for n < l ( η ) and otherwise eval ( η, n ) = 0 . Also add a bijection e : K → M . Adda predicate L for L and a truth valuation T V : L × T → { , } as in Section 3.1 (this timethere is no need to add finite subsets of formulas). Add a function d : N → L mapping n to (cid:8) ψ n (cid:0) v ; v , . . . , v | y n | (cid:1)(cid:9) and a function a : T → T such that if η ∈ λ <ω then a ( η ) ∈ λ | y l ( η ) | , { T V ( d ( i ) , h x i ⌢ a ( η | i )) | i < l ( η ) } is consistent and such that if η, ν ∈ T are incomparable then { T V ( d ( l ( η )) , h x i ⌢ a ( η )) , T V ( d ( l ( ν )) , h x i ⌢ a ( ν )) } is inconsistent. Also add a map k : ω → M such that k ( i ) = b i . Let M = ( M , N , K , T , L ) with this additional structure, and set T = Th ( M ) and L = L ( T ) . In models of T , we will denote by ω the (interpretations of the)standard natural numbers, while elements in N which are not from ω are nonstandard.(*) Note that Lemma 3.2 still holds, with some minor adjustments to the proof (replacing N by K ), giving us a definable function H : K × N → K such that if M | = T and M ↾ L is ℵ -saturated, then for any function f : ω → K M , there is some m f ∈ K M with theproperty that H M ( m f , n ) = f ( n ) for all n ∈ ω .Suppose that M | = T and M ↾ L is κ -saturated. Note that κ ≥ ℵ and hence M ↾ L is ℵ -saturated. Let p ( x ) ∈ S L ( A ) be some complete type where A ⊆ M is of size | A | = κ . Since cf ( κ ) = ω we can write A as an increasing union A = S i<ω A i where | A i | < κ . For i < ω , let c i | = p | A i . By (*), there is some m ∈ K M such that H ( m, i ) = e − ( c i ) for all i < ω . For every ϕ ( x ) ∈ p , there is k ϕ < ω such that the set D ϕ = (cid:8) i ∈ N M (cid:12)(cid:12) ∀ j ∈ [ k ϕ , i ] M | = ϕ ( e ( H ( m, j ))) (cid:9) contains [ k ϕ , ω ) . Note that D ϕ is convex, and that it is A -definable in L . By overspill, D ϕ XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 18 contains some nonstandard element d ϕ ∈ N M . Let C = { d ϕ | ϕ ∈ p } ⊆ N M . Note that if r ∈ N M is nonstandard and r ≤ d ϕ then r ∈ D ϕ , so that if r ≤ d ϕ for all ϕ ∈ p then e ( H ( m, r )) | = p sothat p is realized.Since | A | ≤ κ , | C | ≤ κ , and since κ is singular, the cofinality of C (going down) is < κ . Let C ′ ⊆ C be a coinitial subset of size < κ . To conclude, we show that the set Γ = { x ∈ N } ∪ { n < x | n < ω } ∪ { x ≤ d | d ∈ C ′ } is realized in M . Recall the choice of ϕ and the function k above, and consider the set Π = { ϕ ( k ( i ) , y ) | i < ω }∪{¬ ϕ ( k ( d ) , y ) | d ∈ C ′ } . Then Π is consistent by choice of ϕ , since all elementsin C ′ are nonstandard. By κ -saturation Π is realized, say by f ∈ M | y | . Let g ∈ N M be minimalsuch that M | = ¬ ϕ ( k ( g ) , f ) . Then g is nonstandard and g ≤ d for all d ∈ C ′ (since ¬ ϕ ( k ( d ) , f ) holds for all d ∈ C ′ ), so g | = Γ and we are done. (cid:3) Combining Theorem 4.1 with Theorem 4.8 we get: Corollary 4.9. Let T be unstable theory. Suppose that κ is singular with cofinality ω such that | T | < κ , κ + = 2 κ and (cid:3) κ holds. Then, T is supersimple iff T has PC-exact saturation at κ . Corollary 4.10. Suppose that T is a countable theory. Suppose that κ is singular with cofinality ω such that ℵ < κ , κ + = 2 κ and (cid:3) κ holds. Then T is supersimple iff T has PC-exact saturationat κ .Proof. If T is unstable this follows from Corollary 4.9. If T is stable and supersimple then T issuperstable so T is κ -stable and hence has PC-exact saturation by Coroallry 3.11. On the otherhand, if T has PC-exact saturation at κ then T is κ -stable. But if T is not superstable then bythe stability spectrum theorem [She90, III], κ ℵ = κ , contradicting the cofinality assumption. (cid:3) On local exact saturation and cofinality ω In this section we will see that for κ of cofinality ω , having local PC-exact saturation definesthe class of supershort simple theories: the class of theories for which every local type does notfork over a finite set. Before doing that, we discuss stable theories.5.1. Stable theories. Here we will prove that contrary to the situation with PC-exact saturation(i.e., to Corollary 3.11), stable theories always have local PC-exact saturation. The proof is similarto the proof that λ -stable theories have saturated models of size λ [She90, Theorem III.3.12], buta bit simpler. Definition 5.1. For any theory T , κ loc ( T ) is the smallest cardinal κ such that any local type p ∈ S ∆ ( A ) (see Definition 2.4) does not fork over a set of size < κ . If no such cardinal exists, then κ loc ( T ) = ∞ . XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 19 In other words, the definition is the same as that of κ ( T ) , where types are replaced with localtypes.For stable theories it is always ℵ : Proposition 5.2. Suppose that T is stable. Then κ loc ( T ) = ℵ : every local type p ∈ S ∆ ( A ) doesnot fork over a finite subset A ⊆ A .Proof. Let q ∈ S ( C ) be a global non-forking extension of p over A . Then q is definable over acl eq ( A ) . In particular q ↾ ∆ is definable over acl eq ( A ) for some finite subset A ⊆ A , so q ↾ ∆ does not fork over A . (cid:3) Lemma 5.3. Suppose that T is a stable L -theory. Suppose that h M i | i < µ i is an increasingcontinuous sequence of λ -locally saturated models. Then M µ = S i<µ M i is also λ -locally saturated.Proof. If λ = ℵ , this is clear so assume that λ > ℵ . We are given p ( x ) ∈ S ∆ ( A ) , A ⊆ M µ , | A | < λ , and we want to realize p in M µ ( x is any finite tuple of variables). Let L ′ be a countablesublanguage of L containing ∆ . The models M i ↾ L ′ are still λ -locally saturated, so we mayassume that L = L ′ and in particular it is countable. By Proposition 5.2, p does not fork overa finite set B ⊆ A . In particular, B ⊆ M i for some i < µ . Find a countable model M ′ ≺ M i containing B . Let q be a global extension of p which does not fork over B . By stability, q isdefinable and finitely satisfiable over M ′ .By stability, if h a i | i < ω i is any indiscernible sequence and ϕ ( x, y ) is some formula then forany b , either for almost all i < ω (i.e., all but finitely many) ϕ ( a i , b ) holds or for almost all i < ω , ¬ ϕ ( a i , b ) holds. By compactness, there is some finite set of formulas ∆ and N < ω such that if h a i | i < N i is any ∆ -indiscernible sequence then for any (partition of any) formula ϕ ( x, y ) from ∆ and any b , it cannot be that ϕ ( a i , b ) for i < N and ¬ ϕ ( a i , b ) for N ≤ i < N .As M i is λ -locally saturated (and λ > ℵ ), we can realize in M i a ∆ -Morley sequence generatedby q over M ′ : a | = ( q ↾ ∆ ) | M ′ , a | = ( q ↾ ∆ ) | M ′ a , etc. It is not hard to see that in fact thesequence h a n | n < ω i realizes (cid:0) q ( ω ) | M ′ (cid:1) ↾ ∆ (the latter is just the restriction of q ( ω ) | M ′ to theset of formulas from ∆ in the variables h x n | n < ω i over M ′ where | x n | = | x | ). For a proof, see[KS14, Claim 4.11]. In particular, h a n | n < ω i is a ∆ -indiscernible sequence over M ′ . We cancontinue and realize in M i a ∆ -Morley sequence h a i | i < λ i generated by q over M ′ .We claim that for some i < λ , a i | = p . Indeed, suppose not. This means that for every i < λ , for some ϕ i ( x, y ) from ∆ and some b i ∈ A , ¬ ϕ i ( a i , b i ) holds while ϕ i ( x, b i ) ∈ p . As ∆ is finite and | A | < λ , there are some ϕ and b ∈ A such that ¬ ϕ ( a i , b ) holds for all i ∈ I where I ⊆ λ is infinite and ϕ ( x, b ) ∈ p . However, we can now realize a ′ | = q ↾ ∆ | M ′ b ∪{ a i | i ∈ I } , a ′ | = ( q ↾ ∆) | M ′ b ∪{ a i | i ∈ I }∪ { a ′ } , etc (where a ′ i ∈ C ). Since q extends p , ϕ ( a ′ i , b ) holds for all i < ω ,so we have a contradiction to the choice of ∆ . (cid:3) XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 20 Lemma 5.4. Suppose that T is stable. Then for any M | = T there is an extension M ′ ≻ M suchthat M ′ is locally saturated and | M ′ | ≤ | M | + | T | .Proof. We may assume that | T | ≤ | M | . Let κ = | M | .For every finite set of formulas ∆ , the number of ∆ -types over M is bounded by κ (by stability).Hence there is some M ≻ M such that M realizes every local type in S ∆ ( M ) and | M | = κ .This allows us to construct a continuous increasing elementary chain h M α | α < κ i starting with M = M with the properties that | M α | = κ and for each α < κ , M α +1 realizes every local type in S ∆ ( M α ) . Let M ′ = S α<κ M α .If κ is regular then M ′ is as requested.Otherwise, for every regular λ < κ , M α + λ is λ -locally saturated. Since M ′ = S { M α + λ | α < κ } then by Lemma 5.3, M ′ is λ -locally saturated. Since this is true for every such λ , M ′ is κ -locallysaturated. (cid:3) Theorem 5.5. Suppose that T is stable. Then for every cardinal κ ≥ | T | , T has local PC-exactsaturation at κ .Proof. Suppose that T ⊇ T has cardinality | T | . Without loss of generality assume that T hasSkolem functions. We construct a sequence of T -models h M i | i < ω i and T -models h N i | i < ω i such that: – N i = Sk ( M i ) ; N i ↾ L ≺ M i +1 ; M i is locally saturated and | M i | = κ .For the construction use Lemma 5.4. By Lemma 5.3, M = S i<ω M i is locally saturated, and byconstruction it is in P C ( T , T ) . It is not κ + -locally saturated since | M | = κ (so does not realizethe local type { x = a | a ∈ M } ). (cid:3) Simple theories. In this section we will analyze local PC-exact saturation in the context ofsimple theories. We start with a positive result: Theorem 5.6. Assume that T is a complete simple L -theory, T ⊇ T is a theory in L ⊇ L , | T | ≤ | T | . Also, assume that κ is a singular cardinal such that κ loc ( T ) ≤ µ = cf ( κ ) , | T | < κ , κ + = 2 κ and (cid:3) κ holds. Then P C ( T , T ) has local exact saturation at κ .Proof. The proof is almost exactly the same as the proof of Theorem 4.1 (where we also assumedinstability).By Theorem 5.5, we may assume that T is unstable, so there is an L -formula ϕ ( x, y ) with theindependence property. We find an L -indiscernible sequence I witnessing this and define λ i , I i for i < µ as in the proof of Theorem 4.1. We also define the class C in exactly the same way.The proof of the parallel to Main Lemma 4.3 is similar, but the first step is to say that given alocal type p ( x ) ∈ S ∆ ( C ) , without loss of generality it does not fork over A ∗ = A ∩ C , so we may XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 21 extend it to a complete type p ′ ( x ) ∈ S ( C ) which also does not fork over A ∗ . Then we continuewith the same proof.Note also that x may not be a single variable but we never needed that assumption in the proofof Main Lemma 4.3. (cid:3) We will now discuss κ loc ( T ) (see Definition 5.1), which will lead us to our next result. Claim . For any complete theory T with infinite models, κ loc ( T ) (see Definition 5.1) can beeither ℵ , ℵ or ∞ . In the first two cases T is simple, and in the last case T is not simple. Proof. The proof is standard, but we give details.If T is not simple, then T has the tree property (see [TZ12, Definition 7.2.1]) as witnessedby some formula ϕ ( x, y ) and some k < ω : there is a sequence h a s | s ∈ ω <ω i such that for every s ∈ ω <ω , (cid:8) ϕ (cid:0) x, a s⌢ h i i (cid:1) (cid:12)(cid:12) i < ω (cid:9) is k -inconsistent while for any η ∈ ω ω , Γ η = { ϕ ( x, a η ↾ n ) | n < ω } isconsistent. Let µ be any regular cardinal. By compactness, we may extend the tree to have width λ = (2 µ ) + and height µ (so that s ranges over λ <µ ). For α < µ , find an increasing continuoussequence s α ∈ λ α for α < µ such that s α +1 extends s α and ϕ (cid:0) x, a s α +1 (cid:1) divides (and even k -divides) over (cid:8) ϕ (cid:0) x, a s β (cid:1) (cid:12)(cid:12) β ≤ α (cid:9) (the construction uses the fact that for infinitely many i < λ , a s α ⌢ h i i will have the same type over a s <α ). Letting η = S α<µ s α , any complete ϕ -type extending Γ η over { a s α | α < µ } divides over any subset of size < µ its domain. Since µ was arbitrary, thisshow that κ loc ( T ) = ∞ .Now, if κ loc ( T ) > ℵ , then there is a local type p ∈ S ∆ ( A ) which forks over any countablesubset of A . Let L ′ be a countable subset of the language L of T containing all the symbolsappearing in ∆ . Then any completion q ∈ S L ′ ( A ) of p forks over any countable subset of A ,so T ↾ L ′ does not satisfy local character for non-forking, so it is not simple, and so is T , thus κ loc ( T ) = ∞ .Finally, κ loc ( T ) cannot be any n < ω , since given a , . . . , a n − with a i / ∈ acl ( a = i ) (e.g., a i comefrom an infinite indiscernible sequence), tp = ( a , . . . , a n − /a , . . . , a n − ) divides over any propersubset of { a i | i < n } . (cid:3) Definition 5.8. [CW02, Definition 8] A theory is called supershort if κ loc ( T ) = ℵ . Remark . This is not the precise definition given in [CW02] which is given in terms of dividingchains, but it is equivalent to it: given an infinite dividing chain of conjunctions of a single formula ϕ ( x, y ) as in the definition there, the partial ϕ -type containing them divides over any finite subsetof its domain. On the other hand, if κ loc ( T ) > ℵ and p ∈ S ∆ ( A ) witnesses this (i.e., dividesover every finite A ⊆ A ) for some finite ∆ , then by coding finitely many formulas as one formula(see [She90, Proof of Theorem II.2.12(1)]), we can recover a dividing chain as in the definition in[CW02]. XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 22 Recall that a theory T is low if whenever ϕ ( x, y ) is a formula then there is some n < ω suchthat if h a i | i < ω i is an indiscernible sequence such that { ϕ ( x, a i ) | i < ω } is inconsistent, then itis already n -inconsistent. This is not the original definition from [Bue99, Sha00], which is givenusing local ranks, but it is equivalent to it when T is simple, see [Cas11, Proposition 18.19].The following proposition gives a simple criterion for supershortness. Proposition 5.10. Suppose that T is a simple theory such that if ∆ is a finite set of formulas and p ( x ) ∈ S ∆ ( A ) then there is a finite set of formulas ∆ ′ such that whenever p divides over A ⊆ A ,there is some formula ϕ ( x, y ) from ∆ ′ and some a ∈ C such that p ⊢ ϕ ( x, a ) and ϕ ( x, a ) dividesover A . Then if T is low then κ loc ( T ) = ℵ .Proof. If κ loc ( T ) > ℵ then there is a local type p ( x ) ∈ S ∆ ( A ) (for ∆ finite) such that p dividesover any finite subset of A , in particular A is infinite. Let ∆ ′ be as above. Thus we can constructan increasing chain h A i | i < ω i such that A i ⊆ A are finite and p | A i +1 divides over A i . As ∆ ′ isfinite we can find a single formula ϕ ( x, y ) ∈ ∆ ′ and a i ∈ C such that ϕ ( x, a i ) divides over A i and p | A i +1 ⊢ ϕ ( x, a i ) . If J i is an indiscernible sequence over A i witnessing that ϕ ( x, a i ) divides over A i , by Ramsey and compactness and applying an automorphism we can assume that J i is indis-cernible over a ℵ then there is a formula ϕ ( x, y ) and a sequence of formulas h ψ n ( x, y n ) | n < ω i (where x is a finite tuple of variables and the y n ’s are tuples of variables ofvarying lengths) and a sequence h a η | η ∈ ω <ω i such that: – Each formula ψ n has the form V j Remark 5.9, there is a formula ϕ ( x, y ) and a sequence of formulas h ψ n ( x, y n ) | n < ω i whereeach ψ n is a conjunction of the form V i Let T be any complete theory. Suppose that κ is singular with cofinality ω suchthat | T | < κ , κ + = 2 κ and (cid:3) κ holds. Then, T is supershort iff T has local PC-exact saturation at κ .Proof. Right to left follows from Theorem 5.6. For the other direction, use the same proof as inTheorem 4.8, noting that the proof goes through, because the only use of actual κ -saturation asopposed to local κ -saturation was the use of (*) (i.e., the use of the suitable version of Lemma3.2). Here, all the formulas ψ n are conjunctions of instances of ϕ so the types { ψ i +1 ( x, e ( a ( η i ) (0)) , . . . , e ( a ( η i ) ( | y i +1 | − | i < ω } (using the notation from the proof of Lemma 3.2) are still consistent. Of course, since the x maynow be a tuple of length > , the function H has domain K | x | × N (where x is from ψ n ( x, y n ) ).One more difference is that now the type p ( z ) we wish to realize is in possibly more than onevariable. However this is easy to overcome by taking a tuple of “codes” for the function n c n . (cid:3) Final thoughts and questions NSOP . We would like to extend Theorem 4 to NSOP -theories, but we do not even knowthe situation with elementary classes (i.e., not PC-exact saturation, just exact saturation). Theapproach of mimicking the proof or the proof of [KSS17, Theorem 3.3] using Kim-independenceand all its properties (see [KR19, KR20, KRS19]). However, both proofs use base monotonicityand hence are not applicable. Question 6.1. Is Theorem 4 true for NSOP -theories? Note that [MS17, Theorem 9.30] states that if T has SOP then it has PC-singular compactness (the negation of PC-exact saturation): for some T containing T of cardinality ≤ | T | and everysingular κ > | T | , if M ∈ P C ( T, T ) is κ -saturated then it is κ + -saturated. Thus a positive answerto Question 6.1 will help to “close the gap”.6.2. NIP. In [KSS17, Theorem 4.10] it is proved that if T is NIP, and | T | < κ is singular suchthat κ = κ + , then T has exact saturation at κ iff T is not distal. While the situation for PC-exactsaturation seems less clear, one can ask about local exact saturation (without PC). The proof of the XACT SATURATION IN PSEUDO-ELEMENTARY CLASSES FOR SIMPLE AND STABLE THEORIES 24 direction that if T is distal then T does not have exact saturation at κ goes through in the local case:if T is distal, | T | < κ is singular, then every κ -locally saturated model is κ + -locally saturated. Thisis Proposition [KSS17, Proposition 4.12]. The proof has to be adjusted. Following the notationthere, we elaborate a bit. Given a finite set ∆ and a type p ( x ) ∈ S ∆ ( A ) , let p ′ be an extension of p to S ∆ ′ ( A ) where ∆ ′ = ∆ ∪ { θ ϕ | ϕ ∈ ∆ } (we also consider all possible partitions of formulas in ∆ ).We let b i | = p ′ | A i for i < µ and find d ϕi as there for any ϕ ∈ ∆ . Letting q i = { θ ϕ ( x, d ϕi ) | ϕ ∈ ∆ } (soa finite set), the proof of [KSS17, Proposition 4.13] goes through because p ′ is a complete ∆ ′ -type.We then find e i realizing the ∆ ′′ -type of the finite tuple d i = h d ϕi | ϕ ∈ ∆ i over A i ∪ { b i | i < µ } where ∆ ′′ contains ∆ ′ and the formulas ∀ xθ ϕ ( x, z ) → ϕ ( x, y ) and ∀ xθ ϕ ( x, z ) → ¬ ϕ ( x, y ) for ϕ ∈ ∆ . The rest goes through.However, the other direction, namely that if T is not distal then T has local exact saturationat κ for κ as above seems less clear. The main issue is that the model constructed omits a typeof an element over an indiscernible sequence, and this type is not local. For example if I is anindiscernible set, then the type omitted is that of a new element in the sequence. Question 6.2. Which NIP theories have local exact saturation at singular cardinals as above? References [Bue99] Steven Buechler. Lascar strong types in some simple theories. J. 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