Criteria for exact saturation and singular compactness
CCRITERIA FOR EXACT SATURATION AND SINGULARCOMPACTNESS
ITAY KAPLAN, NICHOLAS RAMSEY, AND SAHARON SHELAH
Abstract.
We introduce the class of unshreddable theories , which containsthe simple and NIP theories, and prove that such theories have exactly satu-rated models in singular cardinals, satisfying certain set-theoretic hypotheses.We also give criteria for a theory to have singular compactness.
Contents
1. Introduction 12. Shredding 33. Respect and exact saturation 104. Examples 145. A criterion for singular compactness 23References 261.
Introduction
The construction of saturated models of a theory T is sensitive to the combinato-rial properties of sets definable in T . Consequently, properties of saturated modelsand their constructions are often reflected in model-theoretic dividing lines, definedin terms of synactic properties of a formula. For example, it is well known that astable theory has a saturated model in every cardinal in which it is stable [13, The-orem III.3.12]. In a similar vein, the third-named author characterized the simpletheories in terms of the saturation spectrum of a theory, namely, the set of car-dinal pairs ( λ, κ ) with λ ≥ κ and every model of size λ extends to a κ -saturatedmodel of the same size [12, Theorem 4.10]. Subsequent work on transferring satu-ration, Keisler’s order, and the interpretability order all suggest that comparisonsbetween saturated models and their constructions yield meaningful measures ofmodel-theoretic complexity [1, 4, 9].A theory T is said to have exact saturation at the cardinal κ if there is a κ -saturated model of T which is not κ + -saturated. If κ is regular and > | T | , every theory has models with exact saturation at κ [8, Theorem 2.4, Fact 2.5], but forsingular κ , this property connects with notions from classification theory. Thesimplest example of a theory without exact saturation at singular κ is the theory of Date : March 3, 2020.This work was supported by the European Research Council grant 338821. Paper no. 1192 inShelah’s publication list. The first author would like to thank the Israel Science Foundation forits support of this research (grants no. 1533/14 and 1254/18). a r X i v : . [ m a t h . L O ] M a r ITAY KAPLAN, NICHOLAS RAMSEY, AND SAHARON SHELAH dense linear orders. Given a singular cardinal κ and a κ -saturated dense linear order I and given any subsets A < B from I with | A | = | B | = κ , there are cofinal andcoinitial subsets A and B of A and B respectively with | A | = | B | < κ . It followsfrom the κ -saturation of I that there is some c ∈ I with a < c < b for all a ∈ A and b ∈ B , hence for all a ∈ A and b ∈ B . By quantifier elimination for the theoryof dense linear orders, it follows that I is κ + -saturated. This example suggests thatfailures of exact saturation are related to the presence of orders. Indeed, it wasshown in [8, Theorem 4.10] that an NIP theory T has exact saturation at a singularcardinal κ if and only if T is not distal (assuming 2 κ = κ + and κ > | T | ).Additionally, [8, Theorem 3.3] showed that if T is simple then T has exactly µ -saturated models for singular µ of cofinality greater than | T | (again assuming2 µ = µ + and, additionally, (cid:3) µ ). In the unstable case, this argument started from awitness ϕ ( x ; y ) to the independence property along an indiscernible sequence I oflength κ and inductively constructed a model M containing I so that every typeover fewer than µ parameters is realized and also so that, for every tuple c from M ,there is an interval from the indiscernible sequence that is indiscernible over c . Thisensures that the model is both µ -saturated yet omits the type { ϕ ( x ; a i ) i even : i ∈ I } .Simplicity theory, via the independence theorem and the forking calculus, playedan important role in that argument.Here, we are interested in both finding criteria for exact saturation in broadermodel-theoretic contexts but also understanding the reach of the argument of [8],which was tailored to simple theories. We introduce shredding , a notion that re-fines forking and exactly captures the obstacle to ensuring that one can realize aformula such that a large interval of a given indiscernible sequence is additionallyindiscernible over the realization. This notion is defined with exact saturation inmind, but it appears to be a fairly fundamental notion and may have uses beyondthe context explored here. We use shredding to define the class of unshreddable the-ories , which are roughly the theories with a bound on the number of times a typecan shred, and observe that both NIP and simple theories are unshreddable. Ourmain theorem is that one may construct exactly saturated models of unshreddabletheories with the independence property for singular cardinals satisfying certainset-theoretic hypotheses. We follow the rough outline of the argument of [8] but, incontrast to the approach taken there, which faced considerable technical issues inadapting the tools of simplicity theory for the construction of an exactly saturatedmodel, our proof, in addition to being more general, is considerably simpler andmore direct.In section 4, we focus on the way that the class of unshreddable theories comparesto other classes from classification theory. We show that there is an unshreddabletheory with SOP , which suggests that the class of unshreddable theories is sub-stantially broader than the simple theories. However, we show subsequently thatneither NSOP nor NTP imply that a theory is unshreddable.In section 5, we consider the dual problem of which conditions on a theory implythe inability to construct exactly saturated models, which we call singular com-pactness . We formulate one such criterion and show that this condition entailsa considerable amount of complexity: theories that meet our condition for everyformula has TP and SOP n for all n . Nonetheless, we show that our condition RITERIA FOR EXACT SATURATION AND SINGULAR COMPACTNESS 3 restricted to a fixed finite set of formulas implies a local version of singular com-pactness. For this local variant, we show that there is an example which satisfiesthe condition for a fixed finite set of formulas which is NSOP .2. Shredding
Basic definitions.
From now on, T will denote a complete first-order theorywith monster model M . In this subsection, we will describe shredding and showthat it can be given a finitary characterization. Definition 2.1.
Let A be a set of parameters and λ an infinite cardinal.(1) We say that ϕ ( x ; a ) λ - shreds over A when there is b such that:(a) b = (cid:104) b α : α < λ (cid:105) is an indiscernible sequence over A .(b) For no α < λ and c ∈ ϕ ( M , a ) is b ≥ α is an indiscernible sequence over Ac .(2) We say a type λ -shreds over A if it implies a formula that λ -shreds over A ,respectively.(3) We say p ∈ S ( B ) λ -shreds over A with a built-in witness if A ⊆ B and anindiscernible sequence witnessing λ -shredding is contained in B .(4) For the above notions, we may omit λ when λ = ( | T | + | A | ) + .(5) We define κ m shred ( T ) to be the minimal regular cardinal κ such that there isno increasing continuous sequence of models (cid:104) M i : i ≤ κ (cid:105) and p ∈ S m ( M κ )so that p (cid:22) M i +1 shreds over M i with a built-in witness, if such a cardi-nal exists. Otherwise, we set κ m shred ( T ) = ∞ . The cardinal κ shred ( T ) =sup m κ m shred ( T ).(6) We say T is unshreddable if κ shred ( T ) < ∞ . Remark . Though we do not use it, it is natural to additionally introduce an asso-ciated notion of forking: say ϕ ( x ; a ) λ -shred-forks over A if ϕ ( x ; a ) (cid:96) (cid:87) i Assume λ = cf ( λ ) > | T | + | A | . The following are equivalent:(1) The formula ϕ ( x ; a ) λ -shreds over A .(2) There are n , b , η , and ψ satisfying:(a) b = (cid:104) b α : α < λ (cid:105) is an A -indiscernible sequence.(b) η = (cid:104) η i : i < k (cid:105) is a finite sequence of increasing functions in n (2 n ) .(c) ψ = (cid:104) ψ l ( x ; y , . . . , y n − ; a (cid:48) l ) : l < k (cid:105) is a sequence of formulas with a (cid:48) l ∈ A .(d) For every δ < λ divisible by n (or just for every limit δ < λ ), we have ϕ ( x ; a ) (cid:96) (cid:95) l Assume λ = cf ( λ ) > | T | + | A | and ϕ ( x ; a ) λ -shreds over A . Thenthere is an A -indiscernible sequence (cid:104) b α : α < λ (cid:105) and m < ω so that • (cid:104) ( b m · α , b m · α +1 , . . . , b m · α + m − ) : α < λ (cid:105) is Aa -indiscernible. • (cid:104) b α : α < λ (cid:105) witnesses that ϕ ( x ; a ) λ -shreds over A and, additionally, for ev-ery c ∈ ϕ ( M , a ) and α < λ , the finite sequence ( b m · α , b m · α +1 , . . . , b m · α + m − ) is not Ac -indiscernible.Proof. Suppose ϕ ( x ; a ) λ -shreds over A . By Lemma 2.3, there is an A -indiscerniblesequence (cid:104) c α : α < λ (cid:105) , a number n < ω , a sequence of L ( A )-formulas ψ = RITERIA FOR EXACT SATURATION AND SINGULAR COMPACTNESS 5 (cid:104) ψ l ( x ; y , . . . , y n − ) : l < k (cid:105) , and a sequence η = (cid:104) η l : l < k (cid:105) with each η l ∈ n (2 n )an increasing function, such that, for every δ < λ divisible by 2 n , ϕ ( x ; a ) (cid:96) (cid:95) l Definition 2.6. For an infinite cardinal λ , we say ϕ ( x ; a ) explicitly λ -shreds over A if there are n , b , η , and ψ satisfying:(1) b = (cid:104) b α : α < λ (cid:105) is an A -indiscernible sequence.(2) η = (cid:104) η l : l < k (cid:105) is a finite sequence of increasing functions in n (2 n ).(3) ψ = (cid:104) ψ l ( x ; y , . . . , y n − ; a (cid:48) l ) : l < k (cid:105) is a sequence of formulas with a (cid:48) l ∈ A .(4) For every δ < λ divisible by 2 n , we have ϕ ( x ; a ) (cid:96) (cid:95) l The following are equivalent:(1) The formula ϕ ( x ; a ) shreds over A .(2) The formula ϕ ( x ; a ) explicitly shreds over A .(3) The formula ϕ ( x ; a ) explicitly ℵ -shreds over A .Proof. (1) = ⇒ (2) is Lemma 2.3, and (2) = ⇒ (3) is immediate, by restricting thewitnessing indiscernible sequence to an initial segment of length ω .(3) = ⇒ (1) Let λ = ( | A | + | T | ) + and suppose ϕ ( x ; a ) explicitly ℵ -shreds, wit-nessed by ( b, n, η, ψ ), where b = (cid:104) b i : i < ω (cid:105) . Define b (cid:48) i = ( b n · i , . . . , b n · i +2 n − )for all i < ω . The sequence (cid:104) b (cid:48) i : i < ω (cid:105) is also A -indiscernible and, without lossof generality, by (the proof of) Corollary 2.5, we may assume further that it is Aa -indiscernible. Then applying compactness, we can stretch it to b (cid:48) = (cid:104) b (cid:48) i : i < λ (cid:105) with b (cid:48) i = ( b n · i , . . . , b n · i +2 n − ) for all i < λ . Then the sequence (cid:104) b i : i < λ (cid:105) is A -indiscernible and, together with n , η , and ψ witnesses that ϕ ( x ; a ) explicitly λ -shreds, hence λ -shreds. This shows (1). (cid:3) ITAY KAPLAN, NICHOLAS RAMSEY, AND SAHARON SHELAH Lemma 2.8. Suppose A is a set of parameters and B ⊆ A . The following areequivalent:(1) ϕ ( x ; a ) shreds over A .(2) There is an A -indiscernible sequence b = (cid:104) b i : i < λ (cid:105) for λ = ( | A | + | T | ) + such that for no c ∈ ϕ ( M ; a ) and for no α < λ is b ≥ α indiscernible over Bc .(3) ϕ ( x ; a ) explicitly shreds over A witnessed by a tuple ( b, n, η, ψ ) , where theformulas ψ have no parameters (i.e. are over the empty set).Proof. (2) = ⇒ (1) is clear by the definition of shredding, since in particular (2)entails that for no c ∈ ϕ ( M ; a ) and α < λ is b ≥ α indiscernible over Ac .(3) = ⇒ (2) since, if b = (cid:104) b α : α < λ (cid:105) , then, for all δ < λ divisible by 2 n , we havethe implication ϕ ( x ; a ) (cid:96) (cid:95) l Suppose B ⊆ A and ϕ ( x ; a ) shreds over A , then ϕ ( x ; a ) shreds over B . Proposition 2.10. Suppose κ is a regular cardinal and m < ω . The following areequivalent:(1) There is an increasing sequence A = (cid:104) A i : i ≤ κ (cid:105) with A κ = (cid:83) i<κ A i and p ∈ S m ( A κ ) and such that p (explicitly) shreds over A i .(2) There is an increasing continuous sequence of models M = (cid:104) M i : i ≤ κ (cid:105) with M κ = (cid:83) i<κ M i and some p ∈ S m ( M κ ) such that p (cid:22) M i +1 shreds over M i with a built-in witness.Proof. The direction (2) = ⇒ (1) is immediate by Lemma 2.7, taking A i = M i forall i ≤ κ .(1) = ⇒ (2): for each i < κ , fix a formula ϕ i ( x ; a i ) ∈ p that explicitly shreds over A i , witnessed by ( b i , n i , η i , ψ i ). By Lemma 2.8, we may assume that b i and ψ i havebeen chosen so that the formulas in ψ i have no parameters. By the regularity of κ , RITERIA FOR EXACT SATURATION AND SINGULAR COMPACTNESS 7 after replacing the sequence with a subsequence, we may assume ϕ ( x ; a i ) ∈ p (cid:22) A i +1 .Moreover, without loss of generality, we may assume b i = (cid:104) b i,j : j < ω (cid:105) for all i < κ .Our assumption that ψ i contains no parameters entails that ϕ ( x ; a i ) explicitlyshreds over any subset of A i and, in particular, that ϕ ( x ; a i ) shreds over a
Here we establish some preliminaryconnections between the concepts of shredding and unshreddable theories with NIPand simplicity. ITAY KAPLAN, NICHOLAS RAMSEY, AND SAHARON SHELAH Definition 2.12. The formula ϕ ( x ; y ) has the independence property if for every n , there are a , . . . , a n − and tuples b w for every w ⊆ { , . . . , n − } so that | = ϕ ( a i , b w ) ⇐⇒ i ∈ w. A theory is said to have the independence property if some formula does modulo T , otherwise T is NIP.Equivalently, the formula ϕ ( x ; y ) has the independence property if there is anindiscernible sequence (cid:104) a i : i < ω (cid:105) and b so that | = ϕ ( a i , b ) if and only if i is even(see, e.g., [15, Lemma 2.7]). Proposition 2.13. If λ = cf ( λ ) > | T | + | A | and some consistent formula ϕ ( x ; a ) λ -shreds over A , then T has the independence property.Proof. Suppose ϕ ( x ; a ) λ -shreds over A . Then by Lemma 2.7, it explicitly λ -shredsso we may fix k , n , ψ , η , and b = (cid:104) b α : α < λ (cid:105) as in the definition of explicitshredding. Let c be an arbitrary element of ϕ ( M ; a ). By the pigeonhole principle,there is a subset X ⊆ λ of size λ , l < k , and t ∈ { , } so that | = ψ l ( c ; b ω · α , . . . , b ω · α + n − , a (cid:48) l ) t ∧ ψ l ( c ; b ω · α + η l (0) , . . . , b ω · α + η l ( n − , a (cid:48) l ) − t for all α ∈ X . Let (cid:104) α i : i < λ (cid:105) be an increasing enumeration of X . For i < λ even, we define d i = ( b ω · α i , . . . , b ω · α i + n − ) and for i < λ odd, we define d i =( b ω · α i + η l (0) , . . . , b ω · α i + η l ( n − ). Then (cid:104) d i : i < λ (cid:105) is an A -indiscernible sequence, bythe A -indiscernibility of b , and we have c | = { ψ l ( x, d i , a (cid:48) l ) t : i < λ even } ∪ { ψ l ( x ; d i , a (cid:48) l ) − t : i < λ odd } , which shows χ ( x, z ; y ) = ψ l ( x, y, z ) has the independence property. (cid:3) Recall that a formula ϕ ( x ; a ) divides over a set A if there is an A -indiscerniblesequence (cid:104) a i : i < ω (cid:105) such that { ϕ ( x ; a i ) : i < ω } is inconsistent. A formula ϕ ( x ; b ) forks over A if ϕ ( x ; b ) (cid:96) (cid:87) i<κ ψ ( x ; a i ) where each ψ i ( x ; a i ) divides over A . A typedivides or forks over A if it implies a formula that respectively divides or forks over A . A theory is called simple if there is a cardinal κ such that, whenever p is a type(in finitely many variables) over A , there is B ⊆ A over which p does not fork with | B | < κ . The least such cardinal κ is called κ ( T ) and the least such regular cardinalis called κ r ( T ). Proposition 2.14. If ϕ ( x ; a ) shreds over A then ϕ ( x ; a ) forks over A .Proof. Suppose λ = ( | T | + | A | ) + and, by Lemma 2.7, we know ϕ ( x ; a ) explicitly λ -shreds over A . Hence, there are is an A -indiscernible sequence b = (cid:104) b i : i < λ (cid:105) such that that there is a sequence of L ( A )-formulas (cid:104) ψ l ( x ; y , . . . , y n − ) : l < k (cid:105) and a sequence (cid:104) η l : l < k (cid:105) with the property that that ϕ ( x ; a ) (cid:96) (cid:95) l Given ( a β ) β<α , to choose a α , first apply Ramsey and compactness to extract from b α a sequence b ∗ α = (cid:104) b ∗ ω · α + i : i < ω (cid:105) which is Aab <α -indiscernible. Then as b α ≡ Ab <α b ∗ α , we can choose a α so that a α b α ≡ Ab <α ab ∗ α . The sequence ( a α ) α<λ satisfies both(1) and (2) by construction. By Ramsey, compactness, and automorphism, we maymoreover assume the sequence (cid:104) ( a α , b α ) : α < λ (cid:105) is an A -indiscernible sequence.By the finite Ramsey theorem, there is n ∗ so that n ∗ → (2 n ) n k . Let Λ = { ν ∈ n ( n ∗ ) : ν increasing } and for ν ∈ Λ, let b α,ν = ( b ω · α + ν ( i ) ) i< n . Let ϕ (cid:48) ( x ; b α,ν )(suppressing parameters from A ) denote the formula (cid:94) l As a corollary, we obtain the following: Proposition 2.15. If T is simple, then κ shred ( T ) ≤ κ r ( T ) .Proof. Suppose not. Let κ = cf( κ ) > | T | and suppose we have the following: • (cid:104) M i : i ≤ κ (cid:105) is an increasing sequence of models of T . • p ( x ) = { ϕ ( x ; a i ) : i < κ } is a consistent partial type. • ϕ ( x ; a i ) shreds over M i . • a i ∈ M i +1 . Then by Proposition 2.14, p forks over M i for all i < κ . Let M κ = (cid:83) i<κ M i . As T is simple, there is subset A ⊆ M κ with | A | < κ r ( T ) such that p does not fork over A . As κ is regular, there is some i < κ so that A ⊆ M i , from which it follows that p does not fork over M i as well, a contradiction to the definition of κ r ( T ). (cid:3) Corollary 2.16. The class of unshreddable theories contains the NIP and simpletheories.Proof. This follows immediately from Proposition 2.13 and Proposition 2.15. (cid:3) Respect and exact saturation Respect. For the entirety of this subsection, we fix a singular cardinal µ .Writing cf( µ ) = κ , we will assume there is an increasing and continuous sequence ofcardinals λ = (cid:104) λ i : i ≤ κ (cid:105) such that λ > κ , λ i +1 is regular for all i < κ , and λ κ = µ .We will assume we have fixed for each i < κ a sequence a i = (cid:104) a i,j : j < λ i +1 (cid:105) ,which is a
Suppose i < κ and A is a set of parameters.(1) We say that A respects a i when for any finite subset C ⊆ A , there is α < λ i +1 such that a i, ≥ α is C -indiscernible.(2) We say p ∈ S <ω ( A ) respects a i when, for every c | = p , the set Ac respects a i . Remark . In Definition 3.1(1), by the regularity of λ i +1 , we could have insteadasked for the existence of such an α < λ i +1 for any C ⊆ A with | C | < λ i +1 , sincethere are fewer than λ i +1 finite subsets of any such C . Definition 3.3. We define K to be the class of A such that:(1) A = (cid:104) A i : i ≤ κ (cid:105) is increasing continuous.(2) | A i | = λ i for all i < κ .(3) a i ⊆ A i +1 for all i < κ .(4) A i respects a i for all i < κ , i.e. there is some α < λ i +1 such that a i, ≥ α is A i -indiscernible, using Remark 3.2.Given A, B ∈ K , we say A ≤ K B if A j ⊆ B j for all j < κ . We say A ≤ K ,i B if A j ⊆ B j for all j satisfying i ≤ j < κ and A ≤ K , ∗ B if A ≤ K ,i B for some i < κ .We may omit the K subscript when it is clear from context. Lemma 3.4. Suppose p is a partial 1-type over A κ with | dom ( p ) | ≤ λ i . Then thereis some i (cid:48) ≥ i and p (cid:48) ⊇ p such that | dom ( p (cid:48) ) | ≤ λ i (cid:48) and, if q is a type over A κ extending p (cid:48) , then q does not shred over A i (cid:48) .Proof. Suppose not. Then we will construct an increasing sequence of types (cid:104) p j : j < κ (cid:105) extending p and an increasing sequence of ordinals (cid:104) i j : j < κ (cid:105) such that | p j | = λ i j and p j shreds over A i j for all j < κ . To begin, we set i = i anduse our assumption to find some p ⊇ p such that p shreds over A i . We mayassume dom( p ) contains A i and has cardinality λ i . Given any (cid:104) p j : j < α (cid:105) and (cid:104) i j : j < α (cid:105) for α ≥ 1, we put p (cid:48) = (cid:83) j<α p j and i (cid:48) = sup j<α i j (here we make useof the fact that κ is regular). Then | p (cid:48) | = λ i (cid:48) and p (cid:48) extends p . Let i α = i (cid:48) + 1. As i α ≥ i , by hypothesis, there is some type p α ⊇ p (cid:48) such that p α shreds over A i (cid:48) +1 . RITERIA FOR EXACT SATURATION AND SINGULAR COMPACTNESS 11 As this will be witnessed by a single formula, we may assume dom( p α ) contains A i α and | p α | = λ i α , completing the induction.Let p ∗ = (cid:83) j<κ p j . Then, by construction, we have p ∗ shreds over A i j for all j < κ . By Proposition 2.10, this contradicts κ ( T ) ≤ κ . (cid:3) Lemma 3.5. If A ∈ K and p is a -type over A κ of cardinality < µ , then there is A (cid:48) ∈ K such that A ≤ K A (cid:48) and some c ∈ A (cid:48) κ realizes p ( x ) .Proof. By Lemma 3.4 and the choice of µ , we may extend p to a type p (cid:48) such that,for some i < κ , | dom( p (cid:48) ) | ≤ λ i and no type extending p (cid:48) over A κ shreds over A i ,and hence does not shred over A i (cid:48) for any i (cid:48) ≥ i by base monotonicity. Withoutloss of generality, we may assume p = p (cid:48) .By induction on j ∈ [ i, κ ], we will define types p j ∈ S ( A j ) so that(1) The types p j are increasing with j .(2) For all j ∈ [ i, κ ), p j ∪ p is consistent.(3) For all j ∈ [ i, κ ), if c | = p j +1 , then for some α < λ j +1 , a j, ≥ α is A j c -indiscernible.Let p i ∈ S ( A i ) be any type consistent with p . Given p j , we note that p ∪ p j extends p and therefore does not explicitly shred over A j . Because | p ∪ p j | < λ j +1 ,by compactness and the fact that A j respects a j , there is a realization c | = p ∪ p j and α < λ j +1 such that a j, ≥ α is A j c -indiscernible. We put p j +1 = tp( c/A j +1 ).Finally, given (cid:104) p j : j ∈ [ i, δ ) (cid:105) for δ limit > i , we set p δ = (cid:83) j ∈ [ i,δ ) p j .Define p κ = (cid:83) j ∈ [ i,κ ) p j . Let c realize p κ and define A ∗ by A ∗ j = A j for all j < i +1and A ∗ j = A j c for all j ≥ i + 1. For all j ∈ [ i, κ ), as c realizes p j +1 , we know there is α < λ j +1 such that a j, ≥ α is cA j -indiscernible. It follows that A ∗ ∈ K , completingthe proof. (cid:3) A one variable theorem.Theorem 3.6. For all m , we have κ m shred ( T ) + ℵ = κ ( T ) + ℵ .Proof. The inequality κ m shred ( T ) ≥ κ ( T ) is clear, so it suffices to show κ ( T )+ ℵ ≥ κ m shred ( T ) + ℵ . Suppose κ ≥ κ ( T ) + ℵ is a regular cardinal, (cid:104) λ i : i < κ (cid:105) is an increasing continuous sequence of cardinals with λ > κ + | T | and λ i +1 regularfor all i < κ . Let µ = sup i<κ λ i .We will prove by induction on m that, if κ < κ m shred ( T ), there is an increasing andcontinuous sequence of sets (cid:104) B i : i ≤ κ (cid:105) and q ( y ) ∈ S ( B κ ) such that q (cid:22) B i +1 shredsover B i . This contradictions our assumption that κ ≥ κ ( T ), by Proposition2.10.When m = 1, we immediately have a contradiction since κ ( T ) ≤ κ <κ ( T ).Suppose it has been proven for m and suppose (cid:104) A i : i ≤ κ (cid:105) is an increasingcontinuous sequence of models with | A i | = λ i and p ( x , . . . , x m ) ∈ S m +1 ( A κ ) is atype such that p (cid:22) A i +1 shreds over A i with a built-in witness b i , witnessed by theformula ϕ i ( x , . . . , x m ; a i ) ∈ p (cid:22) A i +1 . Then because b i is A i -indiscernible, we have (cid:104) A i : i ≤ κ (cid:105) ∈ K in the notation of Subsection 3.1 with the b i playing the role of a i . Let p (cid:48) ( x , . . . , x m ) = { ϕ ( x , . . . , x m ; a i ) : i < κ } and let p (cid:48)(cid:48) ( x m ) be defined by p (cid:48)(cid:48) ( x m ) = ( ∃ x , . . . , x m − ) (cid:94) p (cid:48) ( x , . . . , x m )= { ( ∃ x , . . . , x m − ) (cid:94) ϕ ∈ w ϕ ( x , . . . , x m − ) : w ⊆ p (cid:48) finite } . Note that | p (cid:48)(cid:48) | = κ < µ . By Lemma 3.5, there is B = (cid:104) B i : i ≤ κ (cid:105) ∈ K such A ≤ K B and such that p (cid:48)(cid:48) is realized by some c ∈ B κ . By the definition of K , for each i < κ ,there is some α i < λ i +1 such that b i, ≥ α i is B i -indiscernible. Let i ∗ be minimal suchthat c ∈ B i ∗ and let q ( x , . . . , x m − ) = p (cid:48) ( x , . . . , x m − , c ). Let q (cid:48) ∈ S ( B κ ) be anycompletion of q . Then for all i ≥ i ∗ , we have that q (cid:48) (cid:22) B i +1 shreds over B i withthe built-in witness b i, ≥ α i . Reindexing by setting B (cid:48) i = B i ∗ + i and a i,j = b i,α i + j forall i < κ and j < λ i +1 , we may apply the induction hypothesis to complete theproof. (cid:3) Exact saturation. As in Subsection 3.1, we fix a singular cardinal µ . Writingcf( µ ) = κ , we will assume there is an increasing and continuous sequence of cardinals λ = (cid:104) λ i : i ≤ κ (cid:105) such that λ > κ , λ i +1 is regular for all i < κ , and λ κ = µ .We write I to denote { ( i, α ) : i < κ, α < λ i +1 } ordered lexicographically. Wewrite I i, ≥ β = { ( j, α ) : j = i and α ≥ β } and we write I i for I i, ≥ . We also fix anindiscernible sequence a = (cid:104) a t : t ∈ I (cid:105) . We similarly write a i, ≥ β for (cid:104) a t : t ∈ I i,β (cid:105) and a i for (cid:104) a t : t ∈ I i (cid:105) . If i < κ , and α < β < λ i +1 , we write a i,α,β for the sequence (cid:104) a j,γ : j = i, γ ∈ [ α, β ) (cid:105) . Note that, in particular, we have a i is a
Suppose κ is an uncountable cardinal. For a club C , we writeLim( C ) for the set { α ∈ C : sup( C ∩ α ) = α } . We write (cid:3) κ for the followingassertion: there is a sequence (cid:104) C α : α ∈ Lim( κ + ) (cid:105) such that(1) C α ⊆ α is club.(2) If β ∈ Lim( C α ) then C β = C α ∩ β .(3) If cf( α ) < κ , then | C α | < κ .We call such a sequence a square sequence (for κ ). RITERIA FOR EXACT SATURATION AND SINGULAR COMPACTNESS 13 The following remark was noted in [8, Remark 3.2] —it will play a similar rolein our deduction of the main theorem. Remark . Suppose (cid:104) C α : α ∈ Lim( κ + ) (cid:105) is a square sequence and C (cid:48) α = Lim( C α ).Then we have the following:(1) If C (cid:48) α (cid:54) = ∅ then if sup( C (cid:48) α ) (cid:54) = α then C (cid:48) α has a last element and cf( α ) = ω .If C (cid:48) α = ∅ then cf( α ) = ω .(2) For all β ∈ C (cid:48) α , C (cid:48) β = C (cid:48) α ∩ β .(3) If cf( α ) < κ , then | C (cid:48) α | < κ .The following is the main theorem of the section. The proof follows [8, Theorem3.3]. Theorem 3.9. If T has the independence property and κ shred ( T ) < ∞ , then T hasan exactly µ -saturated model for any singular µ > | T | of cofinality κ ≥ κ shred ( T ) such that (cid:3) µ and µ = µ + .Proof. Let (cid:104) C α : α ∈ Lim( µ + ) (cid:105) be a sequence as in Remark 3.8. Note that, for all α ∈ Lim( µ + ), we have that | C α | < µ by condition (3) of Remark 3.8, as α < µ + and hence cf( α ) < µ . Partition µ + into { S α : α < µ + } so that each S α has size µ + .By induction, we will construct a sequence of pairs (cid:104) ( A α , p α ) : α < µ + (cid:105) such that(1) A α = (cid:104) A α,i : i < κ (cid:105) ∈ K .(2) p α = (cid:104) p α,β : β ∈ S α \ α (cid:105) is an enumeration of all complete 1-types oversubsets of (cid:83) i A α,i of size < µ (using | T | < µ and 2 µ = µ + ).(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, then for any i < κ such that | C α | < λ i and β ∈ C α , then wehave that A β ≤ i A α .At stage 0, we define A to be the minimal sequence in K —that is, A ,i = (cid:83) a i n . Then by choice of i n , A β n ≤ i n A β n +1 so A β n ,j ⊆ A β n +1 ,j . As the sequence (cid:104) i n : n < ω (cid:105) is increasing, we have also j > i n − so, by the inductive hypothesis, A γ,j ⊆ A β n ,j so, by transitivity, A γ,j ⊆ A β n +1 ,j asdesired.Now suppose i < κ , | C α | < λ i , and β ∈ C α . Then β ≤ γ and as A γ ≤ A α we havein particular that A γ ≤ i A α , so we may assume β < γ . Then β ∈ C α ∩ γ = C γ and | C γ | = | C α ∩ γ | < λ i so it follows by induction that A β ≤ i A γ ≤ A α so A β ≤ i A α .To conclude, we define a model M by M = (cid:91) α<κ + i<µ A α,i . By (4), the model M is µ -saturated. Moreover M is not µ + -saturated, as the partialtype { ϕ ( x ; a i,α ) : i < κ, α even } ∪ {¬ ϕ ( x ; a i,α ) : i < κ, α odd } is omitted by (1). (cid:3) Question 3.10. Suppose T is NTP and has the independence property, and as-sume µ is a singular cardinal such that cf( µ ) > | T | , µ = µ + , and (cid:3) µ . Does T have an exactly µ -saturated model? Examples Standard examples for the SOP n hierarchy. Recall the definition of theSOP n heirarchy: Definition 4.1. Suppose n ≥ 3. The theory T has the n th strong order property (SOP n ) if there is a formula ϕ ( x ; y ) and a sequence of tuples (cid:104) a i : i < ω (cid:105) so that(1) | = ϕ ( a i ; a j ) if and only if i < j .(2) { ϕ ( x i , x i +1 ) : i < n − } ∪ { ϕ ( x n − , x ) } is inconsistent.If T does not have SOP n , we say T is NSOP n .By a directed graph we mean a set with a binary relation that is assymetric andirreflexive. Given a natural number n ≥ 3, we let L n = { R ( x, y ) } ∪ { S l ( x, y ) :1 ≤ l < n } be a language with n binary relations. The theory T n is the L n -theoryof directed graphs with no cycle of length ≤ n , where R ( x, y ) is the (assymetric)edge relation and S l ( x, y ) means that there is no directed path of length ≤ l from x to y . More precisely, T n consists of the following axioms: • R ( x, y ) is an irreflexive assymetric relation:( ∀ x, y )[ R ( x, y ) → ¬ R ( y, x )] . RITERIA FOR EXACT SATURATION AND SINGULAR COMPACTNESS 15 • There are no directed loops of length ≤ n . That is, for all k with 1 ≤ k ≤ n ,we have ¬ ( ∃ z , . . . , z k ) (cid:34) (cid:94) i Note that T eliminates quantifiers in the language containing only R , since R ( x, y ) is definable by the formula x (cid:54) = y ∧ ¬ R ( y, x ). For simplicity, we will write R for R . Because algebraic closure in T is trivial, by replacing c by somethingwith the same type over Aa , we may assume c is disjoint from Aab . Define a model M | = T as follows with underlying set Aabc by defining R M = R M (cid:22) Aab ∪ R M (cid:22) Aac. We claim that M | = T . To see this, suppose not and there are distinct d , d , d ∈ M so that R M ( d , d ), R M ( d , d ), and R M ( d , d ). Since M has no directedcycles of length 3, it is impossible for d , d , d to be all contained in Aab or allcontained in Aac . Therefore, without loss of generality, d ∈ Aab \ Aac . But thensince R M ( d , d ) and R M ( d , d ), we have d , d ∈ Aab , by the definition of R M , acontradiction. This shows M has no directed cycle of length 3 so M | = T .Embed M into M over Aab and let c (cid:48) be the image of c . By quantifier elim-ination, we have c (cid:48) ≡ Aa c and, because c (cid:48) is disjoint from Aab , we have b is Ac (cid:48) -indiscernible. (cid:3) Proposition 4.4. κ shred ( T ) = ℵ .Proof. First, we will argue that κ shred ( T ) ≥ ℵ . For each i < ω , find a i and b i = (cid:104) b i,j : j < ω (cid:105) such that b i is a
2. Towards contradiction, suppose A isa set of parameters, ϕ ( x ; a ) shreds over A witnessed by b , ϕ ( x ; a ) shreds over Aa witnessed by b , and { ϕ ( x ; a ) , ϕ ( x ; a ) } is consistent. Because ϕ ( x ; a ) hasno realization c such that b is indiscernible over Ac , it follows by Lemma 4.3 thatany realization of ϕ ( x ; a ) is contained in Aa . Then let c | = { ϕ ( x ; a ) , ϕ ( x ; a ) } .Because c is an element of Aa , it follows that b is Aa c -indiscernible, contradict-ing the fact that b witnesses that ϕ ( x ; a ) shreds over Aa . This completes theproof. (cid:3) An NSOP theory with κ shred ( T ) = ∞ . There is a theory of independencefor NSOP theories that indicates this class of theories may be considered quiteclose to the class of simple theories (see, e.g., [6]). In the next example, however, weshow that within the class of NSOP theories, it is still possible that κ shred ( T ) = ∞ .Recall the definition of SOP : Definition 4.5. A formula ϕ ( x ; y ) is said to have SOP if there is a tree of tuples( a η ) η ∈ <ω satisfying the following:(1) For all η ∈ ω , { ϕ ( x ; a η | α ) : α < ω } is consistent.(2) For all η ⊥ ν in 2 <ω , if ( η ∧ ν ) (cid:95) (cid:69) η and ( η ∧ ν ) (cid:95) ν , then { ϕ ( x ; a η ) , ϕ ( x ; a ν ) } is inconsistent.A theory T is said to have SOP if some ϕ ( x ; y ) has SOP modulo T , otherwise T is NSOP . RITERIA FOR EXACT SATURATION AND SINGULAR COMPACTNESS 17 The following theory is a variation on the generic theory of selector functions T ∗ considered in [6, Subsection 9.2]. The language L for our example consists of unarypredicates F, O , O , and O , binary relations E, R , and R , and a binary functioneval. The theory T consists of the following axioms:(1) F , O , and O partition the universe and O = O ∪ O .(2) E ⊆ O is an equivalence relation.(3) eval : F × O → O is a selector function:(a) ( ∀ x ∈ F )( ∀ y ∈ O ) [ E ( y, eval( x, y ))].(b) ( ∀ x ∈ F )( ∀ y, z ∈ O ) [ E ( y, z ) → eval( x, y ) = eval( x, z )].(4) The relations R , R satisfy:(a) R ⊆ O × O .(b) R ⊆ F × O .(c) ( ∀ x ∈ F )( ∀ z ∈ O ) [ R (eval( x, z ) , z ) ↔ R ( x, z )].Define K to be the class of finite models of T . Lemma 4.6. The class K is a Fra¨ıss´e class. Moreover, it is uniformly locally finite.Proof. HP is clear as the axioms of T are universal. The argument for JEP isidentical to that for SAP, so we show SAP. Suppose A, B, C ∈ K where A ⊆ B, C and B ∩ C = A . It suffices to define a L -structure with domain D = B ∪ C ,extending both B and C . First, note that if F B is non-empty, then every E B -class intersects O B , but if F B = ∅ , it is possible that there are E B -equivalenceclasses disjoint from O B . In this latter case, we can extend B to B (cid:48) so that eachequivalence class contains an element of O : Let ( K i ) i Define a ternary relation | (cid:94) ∗ on small subsets of M by: a | (cid:94) ∗ C b if and only if(1) dcl( aC ) /E ∩ dcl( bC ) /E ⊆ dcl( C ) /E .(2) dcl( aC ) ∩ dcl( bC ) ⊆ dcl( C ). RITERIA FOR EXACT SATURATION AND SINGULAR COMPACTNESS 19 where X/E = { [ x ] E : x ∈ X } denotes the collection of E -classes represented by anelement of X . Lemma 4.10. The relation | (cid:94) ∗ satisfies the independence theorem over models: if M | = T ∗ , a ≡ M a (cid:48) , and, additionally, a | (cid:94) ∗ M B , a (cid:48) | (cid:94) ∗ M C and B | (cid:94) ∗ M C then thereis a (cid:48)(cid:48) with a (cid:48)(cid:48) ≡ MB a , a (cid:48)(cid:48) ≡ MC a (cid:48) , and a (cid:48)(cid:48) | (cid:94) ∗ M BC .Proof. Without loss of generality, we may assume that M ⊆ B, C , and that B and C are definably closed. Write a = ( d , . . . , d k − , e , . . . , e l − , f , . . . , f m − ) with d i ∈ F , e j ∈ O , f k ∈ O , and likewise a (cid:48) = ( d (cid:48) , . . . , d (cid:48) k − , e (cid:48) , . . . , e (cid:48) l − , f (cid:48) , . . . , f (cid:48) m − ).Fix an automorphism σ ∈ Aut( M /M ) with σ ( a ) = a (cid:48) . Let U = { u g : g ∈ dcl( aB ) \ B } and V = { v g : g ∈ dcl( a (cid:48) C ) \ C } denote collection of new formal elements with u g = v σ ( g ) for all g ∈ (cid:104) aM (cid:105) \ B . Let, then, a ∗ be defined as follows: a ∗ = ( u d , . . . , u d k − , u e , . . . , u e l − , u f , . . . , u f m − )= ( v d (cid:48) , . . . , v d (cid:48) k − , v e (cid:48) , . . . , v e (cid:48) l − , v f (cid:48) , . . . , v f (cid:48) m − ) . We will construct by hand an L -structure D extending (cid:104) BC (cid:105) with domain U V (cid:104) BC (cid:105) in which a ∗ ≡ B a , a ∗ ≡ C a (cid:48) and a ∗ | (cid:94) ∗ M BC .There is a bijection ι : dcl( aB ) → BU given by ι ( b ) = b for all b ∈ B and ι ( g ) = u g for all g ∈ dcl( aB ) \ B . Likewise, we have a bijection ι : dcl( a (cid:48) C ) → CV given by ι ( c ) = c for all c ∈ C and ι ( g ) = v g for all g ∈ dcl( a (cid:48) C ) \ C . The union ofthe images of these functions is the domain of the structure D to be constructed andtheir intersection is ι ( (cid:104) aM (cid:105) ) = ι ( (cid:104) a (cid:48) M (cid:105) ). Consider BU and CV as L -structures bypushing forward the structure on dcl( aB ) and dcl( a (cid:48) C ) along ι and ι , respectively.Note that ι | (cid:104) aM (cid:105) = ( ι ◦ σ ) | (cid:104) aM (cid:105) .We are left to show that we can define an L -structure on U V (cid:104) BC (cid:105) extendingthat of BU , CV , and (cid:104) BC (cid:105) in such a way as to obtain a model of T . To begin,interpret the predicates by O Di = O BUi ∪ O CVi ∪ O (cid:104) BC (cid:105) i for i = 0 , O D = O D ∪ O D , F D = F BU ∪ F CV ∪ F (cid:104) BC (cid:105) , and R D = R BU ∪ R CV ∪ R (cid:104) BC (cid:105) . Let E D be defined to bethe equivalence relation generated by E BU , E CV , and E (cid:104) BC (cid:105) . The interpretationof the predicates defines extensions of the given structures since if g is an elementof ι ( (cid:104) aM (cid:105) ) = ι ( (cid:104) a (cid:48) M (cid:105) ) then ι − ( g ) is in the predicate O if and only if ι − ( g ) isas well, and, moreover, it is easy to check that our assumptions on a, a (cid:48) , B, C entailthat no pair of inequivalent elements in BU , CV , or (cid:104) BC (cid:105) become equivalent in D .Next we define the function eval D extending eval BU ∪ eval CV ∪ eval (cid:104) BC (cid:105) . We firstclaim that eval BU ∪ eval CV ∪ eval (cid:104) BC (cid:105) is a function. The intersection of the domainsof the first two functions is ι ( (cid:104) aM (cid:105) ) = ι ( (cid:104) aM (cid:105) ). If b, b (cid:48) are in this intersection,we must show eval BU ( b, b (cid:48) ) = c ⇐⇒ eval CV ( b, b (cid:48) ) = c. Choose b , b (cid:48) , c ∈ (cid:104) aM (cid:105) and b , b (cid:48) , c ∈ (cid:104) a (cid:48) M (cid:105) with ι i ( b i , b (cid:48) i , c i ) = ( b, b (cid:48) , c ) for i = 0 , 1. Then since ι = ι ◦ σ on (cid:104) aM (cid:105) , we have M | = eval( b , b (cid:48) ) = c ⇐⇒ M | = eval( σ ( b ) , σ ( b (cid:48) )) = σ ( c ) ⇐⇒ M | = eval( b , b (cid:48) ) = c . Since eval BU and eval CV are defined by pushing forward the structure on (cid:104) aB (cid:105) and (cid:104) a (cid:48) C (cid:105) along ι and ι , respectively, this shows that eval BU ∪ eval CV defines afunction. Now the intersection of (cid:104) BC (cid:105) with BU ∪ CV is BC and, by construction,all 3 functions agree on this set. So the union defines a function. Note that because BU , CV , and (cid:104) BC (cid:105) all contain a model M and therefore havenon-empty F -sort, every E D class is represented by an element of O D . Choose acomplete set of E D -class representatives { d i : i < α } so that if d i represents an E D -class that meets M then d i ∈ M and d i ∈ O . If e ∈ O D is E D -equivalent tosome e (cid:48) and ( f, e (cid:48) ) is in the domain of eval BU ∪ eval CV ∪ eval (cid:104) BC (cid:105) , define eval D ( f, e )to be the value that this function takes on ( f, e (cid:48) ). On the other hand, if f ∈ F D \ ( F BU ∪ F CV ∪ F (cid:104) BC (cid:105) ) or e is not E D -equivalent to any element on whicheval D ( f, − ) has already been defined, put eval D ( f, e ) = d i for the unique d i which is E D -equivalent to e . This now defines eval D on all of F D × O D and, by construction,eval D ( f, − ) is a selector function for E D for all f ∈ F D .To conclude, we must interpret R on D . In order to build a structure thatsatisfies axiom (4), we are forced to interpret R D = { ( f, b ) ∈ F × O : (eval D ( f, b ) , b ) ∈ R D } . In order to ensure that D is an extension of BU , CV , and (cid:104) BC (cid:105) , we have showthat for all X ∈ { BU, CV, (cid:104) BC (cid:105)} , R D (cid:22) X = R X . Suppose we have f, a, b ∈ X witheval X ( f, b ) = a . Then because X is a model of T , we have R X ( a, b ) ⇐⇒ R X ( f, b )and, by construction, R X ( a, b ) ⇐⇒ R D ( a, b ). By definition, R D ( f, b ) ⇐⇒ R D ( a, b ). This shows R D ( f, b ) ⇐⇒ R X ( f, b ), hence R D (cid:22) X = R X .We have already argued that BU and CV are substructures of D - it follows thatevery E D -class represented by an element of a ∗ can only be equivalent to an elementof B or C if it is equivalent to an element of M . Moreover, our construction hasguaranteed that (cid:104) a ∗ M (cid:105) D ∩ (cid:104) BC (cid:105) ⊆ BU ∩ (cid:104) BC (cid:105) D ⊆ B and, by similar reasoning, (cid:104) a ∗ M (cid:105) ∩ (cid:104) BC (cid:105) ⊆ C . This implies (cid:104) a ∗ M (cid:105) D ∩ (cid:104) BC (cid:105) B ∩ C ⊆ M , so a ∗ | (cid:94) ∗ M BC .Embedding D into M over (cid:104) BC (cid:105) , we conclude. (cid:3) Corollary 4.11. The theory T ∗ is NSOP .Proof. The relation | (cid:94) ∗ is easily seen to satisfy properties (1) through (4) fromFact 4.8 and the independence theorem is established in Lemma 4.10. This implies T ∗ is NSOP . (cid:3) Remark . One may additionally show that | (cid:94) ∗ = | (cid:94) K over models. As wewon’t need Kim-independence in what follows, we omit the proof. Proposition 4.13. κ shred ( T ∗ ) = ∞ .Proof. Let κ be an arbitrary regular cardinal. Inductively, we may choose a se-quence of elements (cid:104) a i : i < κ (cid:105) and a sequence of sequences (cid:104) b i : i < κ (cid:105) so that(1) For all i < κ , a i ∈ O .(2) For all i < κ , b i = (cid:104) b i,j : j < ω (cid:105) is an a
An NTP example. In this subsection, we describe an NTP example with κ shred ( T ) = ∞ . Recall the definition of NTP theories: Definition 4.14. A formula ϕ ( x ; y ) has the tree property of the second kind (TP )if there is an array of tuples ( a i,j ) i,j<ω and k < ω satisfying the following:(1) For all f : ω → ω , { ϕ ( x ; a i,f ( i ) ) : i < ω } is consistent.(2) For all i < ω , { ϕ ( x ; a i,j ) : j < ω } is k -inconsistent.A theory is said to have TP if some formula has TP modulo T and is otherwisecalled NTP .The class of NTP contains both the NIP and simple theories, so it is natural toask if NTP implies κ shred ( T ) < ∞ but we show this is not the case.The following fact will be useful in checking that the theory we construct isNTP : Fact 4.15. (1) If T has TP , there is a formula ϕ ( x ; y ) witnessing this with l ( x ) = 1 [2, Corollary 2.9].(2) If ϕ ( x ; y ) has TP , then this will be witnessed with respect to an arrayof parameters ( a i,j ) i,j<ω that is mutually indiscernible —that is, a i is a (cid:54) = i -indiscernible for all i < ω [2, Lemma 2.2].Let L a language consisting of two binary relations R, (cid:69) , and a binary function ∧ and the sublanguage consisting of just (cid:69) and ∧ is L tr . The class K will consistof finite L -structures ( A, (cid:69) A , ∧ A , R A ) so that ( A, (cid:69) A , ∧ A ) is a meet-tree where ∧ A is the meet function, and R A is a graph on A . Denote the class of finite ∧ -trees( A, (cid:69) A , ∧ A ) by K . This is a Fra¨ıss´e class with the strong amalgamation property(SAP) and the theory T tr of its Fra¨ıss´e limit is dp-minimal [15, Exercise 2.50,Example 4.28], which means given a mutually indiscernible array ( a i,j ) i< ,j<ω andelement c , there is some i < a i is c -indiscernible. Lemma 4.16. The class K is a Fra¨ıss´e class. Moreover, the reduct of the Fra¨ıss´elimit of K to L tr is the Fra¨ıss´e limit of K .Proof. HP is clear and JEP will follow from a similar argument to SAP, so we willprove SAP. Fix ˜ A, ˜ B , ˜ B ∈ K such that ˜ A is an L -substructure of both ˜ B and ˜ B and ˜ B ∩ ˜ B = ˜ A . Let A = ˜ A (cid:22) L tr and B i = ˜ B i (cid:22) L tr for i = 0 , 1. By SAP in K ,there is D ∈ F extending both B and B . We may expand D to an L -structure ˜ D by setting R ˜ D = R ˜ B ∪ R ˜ B . This establishes SAP for K .Next, suppose A, B ∈ K and π : A → B is an L tr -embedding. If ˜ A ∈ K isan expansion of A , then we can expand B to the L -structure ˜ B in which R ˜ B = { ( π ( a ) , π ( a (cid:48) )) : ( a, a (cid:48) ) ∈ R ˜ A } . Clearly we have ˜ B ∈ K and π is also an L -embeddingso by [10, Lemma 2.8], the reduct of the Fra¨ıss´e limit of K is the Fra¨ıss´e limit of K . (cid:3) By Lemma 4.16, we know that K has a Fra¨ıss´e limit which is an ω -categoricalexpansion of T tr by a (random) graph. Let T denote its theory and let M and M tr denote the monster models of T and T tr respectively. Lemma 4.17. Suppose we are given an L -indiscernible sequence I = (cid:104) a i : i ∈ Z (cid:105) and an element b so that I is L tr -indiscernible over b . Then there is b (cid:48) ≡ La b sothat I is L -indiscernible over b (cid:48) . Proof. Let σ ∈ Aut L tr ( M /b ) be an automorphism so that σ ( a i ) = a i +1 . Let B denote the L -structure generated by (cid:104) a i : i ∈ Z (cid:105) and let A be the L -structuregenerated by a b . Now expand the L tr -structure (cid:104) b ( a i ) i ∈ Z (cid:105) L tr to an L -structure M by setting R M = R B ∪ (cid:91) i ∈ Z σ i ( R A ) . Claim 1 : If i ∈ Z and c, d ∈ B ∩ σ i ( A ), then ( c, d ) ∈ R B if and only if ( c, d ) ∈ σ i ( R A ). Proof of claim : This is clear if i = 0, since R B = R M (cid:22) B and R A = R M (cid:22) A .In general, if c, d ∈ B ∩ σ i ( A ), there are L tr -terms t, t (cid:48) , s, s (cid:48) so that c = t ( a i ) = t (cid:48) ( b, a i ) d = s ( a i ) = s (cid:48) ( b, a i ) . By indiscernibility, it follows that if σ i ( c (cid:48) , d (cid:48) ) = ( c, d ), then we have c (cid:48) = t ( a < , a , a > ) = t (cid:48) ( b, a ) d (cid:48) = s ( a < , a , a > ) = s (cid:48) ( b, a ) , and we know that ( c (cid:48) , d (cid:48) ) ∈ R B if and only if ( c (cid:48) , d (cid:48) ) ∈ R A , by the i = 0 case. Byindiscernibility, ( c (cid:48) , d (cid:48) ) ∈ R B if and only if ( c, d ) ∈ R B and hence ( c, d ) ∈ R B if andonly if ( c, d ) ∈ σ i ( R A ). (cid:3) Claim 2 : If i > c, d ∈ A ∩ σ i ( A ) then ( c, d ) ∈ R A if and only if ( c, d ) ∈ σ i ( R A ). Proof of claim : As in the proof of the previous claim, there are L tr -terms t, t (cid:48) , s ,and s (cid:48) so that we have the following equalities: c = t ( a , b ) = t (cid:48) ( a i , b ) d = s ( a , b ) = s (cid:48) ( a i , b ) . Then by L tr -indiscernibility over b , we have also t ( a , b ) = t (cid:48) ( a i +1 , b ) and t ( a , b ) = t (cid:48) ( a i +1 , b ), hence t ( a , b ) = t ( a , b ). Likewise, we have s ( a , b ) = s ( a , b ). In partic-ular, this shows σ ( c, d ) = ( c, d ) so the claim follows. (cid:3) Now, by Claim 1, it follows that for all c, d ∈ B , we have ( c, d ) ∈ R M if andonly if ( c, d ) ∈ R B , so M extends B . Likewise, by Claim 2, M extends A and σ i induces an L -isomorphism of A and the structure generated by ba i in M , forall i ∈ Z . Embed M into M over B and let b (cid:48) be the image of b under thisembedding. Then by quantifier-elimination, a b ≡ a i b (cid:48) for all i ∈ Z . After applyingRamsey, compactness, and an automorphism, we can find b (cid:48)(cid:48) ≡ a b (cid:48) so that I is L -indiscernible over b (cid:48)(cid:48) , completing the proof. (cid:3) Corollary 4.18. The theory T is NTP (and is, in fact, inp-minimal).Proof. If T has TP , then, by Fact 4.15 and compactness, there is an L -formula ϕ ( x ; y ) with l ( x ) = 1 that witnesses TP with repect to the mutually indiscerniblearray ( a i,j ) i<ω,j ∈ Z . Let b | = { ϕ ( x ; a i, ) : i < ω } . As T tr is dp-minimal, there is a row i = 0 or i = 1 so that (cid:104) a i,j : j ∈ Z (cid:105) is b -indiscernible in the language L tr . By Lemma4.17, there is b (cid:48) ≡ La i, b such that (cid:104) a i,j : j ∈ Z (cid:105) is b (cid:48) -indiscernible in the language L .Then b (cid:48) | = { ϕ ( x ; a i,j ) : j ∈ Z } , contradicting the row-wise inconsistency requiredfor TP . (cid:3) Proposition 4.19. κ shred ( T ) = ∞ . RITERIA FOR EXACT SATURATION AND SINGULAR COMPACTNESS 23 Proof. Let κ be an arbitrary regular cardinal. Inductively, we may choose a se-quence of elements (cid:104) a i : i < κ (cid:105) and a sequence of sequences (cid:104) b i : i < κ (cid:105) so that(1) For all i < κ , b i = (cid:104) b i,j : j < ω (cid:105) is an a
Illustration of the choice of a i and b i Let p ( x ) = { x (cid:68) a i : i < κ } . Notice that if x (cid:68) a i , then x ∧ b i,j = a i ∧ b i,j andhence x (cid:68) a i (cid:96) R ( x ∧ b i,j , b i,j ) if and only if j is even. It follows that the formula x (cid:68) a i explicitly shreds over a
In this section, we give a sufficient condition for having singular compactness,which is the negation of exact saturation (Definition 5.1 below). If ∆ ( x, y ) is aset of formulas then a (partial) ∆ -type is a consistent set of instances of formulasfrom ∆. We may refer to a { ϕ } -type as a ϕ -type. It is important to note that bya ϕ -type we mean a consistent set of positive instances of ϕ , and do not includeinstances of ¬ ϕ . Definition 5.1. Suppose that T is a complete first order theory and ∆ is a setof formulas. Say that T has singular compactness for ∆ if whenever M | = T is µ -saturated for a singular cardinal µ > | T | then M is µ + , ∆-saturated: for every∆-type p over a set A ⊆ M with | A | ≤ µ , p is realized in M . Condition 5.2. For every formula ϕ ( x, y ) (perhaps in a fixed set of formulas ∆)there is some formula θ ϕ ( x, z ) such that for any finite ϕ -type r ( x ) over M | = T and every finite set A ⊆ M x of realizations of r there is some b ∈ M z such that θ ϕ ( A, b ) holds (i.e., M | = θ ϕ ( a, b ) for all a ∈ A ) and θ ϕ ( x, b ) (cid:96) r ( x ). Lemma 5.3. Suppose that T is a complete first order theory and that Condition5.2 holds for ∆ ( x, y ) . Then T has singular compactness for ∆ .Proof. Let p be a ∆-type over a set A with | A | = µ . Write A = (cid:83) i<κ A i with | A i | < µ , κ < µ . For each i < κ find b i ∈ M such that b i | = p | A i (exists by µ -saturation).By compactness and Condition 5.2, for each ϕ ∈ ∆ find e ϕi ∈ M z such that θ ϕ ( b j , e ϕi ) holds for all j ≥ i and θ ϕ ( x, e ϕi ) (cid:96) tp + ϕ ( b i /A i ), the (positive) ϕ -type of b i over A i . By µ -saturation, find d ϕi ∈ M such that d ϕi ≡ A i ∪{ b i : i<κ } e ϕi . Then { θ ϕ ( x, d ϕi ) : i < κ, ϕ ∈ ∆ } is a type and hence realized in M . (cid:3) When does Condition 5.2 hold? If T is complicated enough, e.g., T = P A or T = ZF C , then it holds since given ϕ ( x, y ), we can choose θ ϕ ( x, z ) = x ∈ z .Indeed, this condition implies that the theory cannot be too tame. Proposition 5.4. Assume T has infinite models. If Condition 5.2 holds for everyformula with one variable x then T has TP , and has SOP n for all n .Proof. We start by showing that T has TP . Let ϕ ( x, z ) be θ x (cid:54) = y ( x, z ). Let ψ ( x, w ) = θ ¬ ϕ ( x, w ). We will show that ξ ( x, zw ) = ϕ ( x, z ) ∧ ψ ( x, w ) witnessesTP . Let { a i : i < ω } be some infinite set in M . Suppose that F is a familyof pairwise disjoint subsets of ω . It is enough to find some b s ∈ M w for every s ∈ F such that ξ ( a i , b s ) holds whenever i ∈ s and { ξ ( x, b s ) , ξ ( x, b t ) } is inconsis-tent (see [5, Lemma 2.19]). By compactness we may assume that F is finite andconsists of finite sets and replace ω by some n < ω .By choice of ϕ ( x, z ) there are c s for s ∈ F such that ϕ ( a i , c s ) holds iff i ∈ s :take the finite type r s = { x (cid:54) = a i : i / ∈ s } and A s = { a i : i ∈ s } and apply Condition5.2. This already shows that T has the independence property so is not NIP.Choose d s similarly by applying Condition 5.2 for ϕ and taking r s = {¬ ϕ ( x, c t ) : t (cid:54) = s, t ∈ F} and A s = { a i : i ∈ s } . Then obviously ξ ( a i , c s d s ) holds if i ∈ s . Also,as ψ ( x, d s ) (cid:96) ¬ ϕ ( x, c t ) for t (cid:54) = s , we are done.Next we show that T has SOP n for all n < ω . As SOP n +1 implies SOP n for all n , it suffices to show T has SOP n for n ≥ ϕ ( x, y ) = θ (cid:54) = ( x, y ), ϕ ( x, y ) = θ ϕ ( x, y ) and in general ϕ n +1 ( x, y n +1 ) = θ ϕ n ( x, y n +1 ). Fix some n < ω . Let χ ( y , . . . , y n − , x ; z , . . . , z n − ; x (cid:48) ) with | z i | = | y i | say that ( ∀ x )[ ϕ i +1 ( x, y i +1 ) → ϕ i ( x, z i )]for all i < n − ϕ n − ( x , y n − ) ∧ ¬ ϕ ( x (cid:48) , y ). We will show that χ witnessesSOP n for all n ≥ (cid:104) a t : t < ω (cid:105) be some infinite sequence in M . For t, i < n , let b it ∈ M y i besuch that ϕ i (cid:0) a s , b it (cid:1) holds iff s ≤ t (i.e., witnessing that ϕ i has the order prop-erty) and ( ∀ x )[ ϕ i +1 (cid:0) x, b i +1 t (cid:1) → ϕ i (cid:0) x, b it (cid:48) (cid:1) ] for all t (cid:48) ≥ t . We may find such b it ’sby induction on i < n using Condition 5.2 and compactness as above. For k < ω ,let ¯ b k = b k . . . b n − k a k . We have that for k, l < ω , M | = χ (cid:0) ¯ b k , ¯ b l (cid:1) if and only if k < l . However, it is impossible that { χ (¯ x k , ¯ x k +1 ) : k < n − } ∪ { χ (¯ x n − , ¯ x ) } is consistent, since if it were realized by ¯ c k = c k . . . c n − k d k for k < n , then ϕ n − (cid:0) d , c n − (cid:1) ⇒ · · · ⇒ ϕ (cid:0) d , c n − (cid:1) but as χ (¯ c n − , ¯ c ) holds, we have that ¬ ϕ (cid:0) d , c n − (cid:1) holds as well which is a contradiction. (cid:3) We give an example where this criterion holds. Example 5.5. Let L = { P i : i < } ∪ { R , , R , , R , } where the P i s are unarypredicates and the R i,j s are binary relation symbols. Let T ∀ say that (cid:104) P i : i < (cid:105) are disjoint and their union covers the universe, that R i,j ⊆ P i × P j and that: (cid:11) If R , ( b, c ) then ( ∀ x ) [ R , ( x, b ) → R , ( x, c )]. Claim . T ∀ is universal, it has the amalgamation property (AP) and the jointembedding property (JEP). RITERIA FOR EXACT SATURATION AND SINGULAR COMPACTNESS 25 Proof. The fact that T ∀ is universal is clear.JEP: suppose that M , M | = T ∀ are disjoint. Let M be the following structure.As a set it is M ∪ M . For every relation symbol Q ∈ L , let Q M = Q M ∪ Q M .AP: suppose that M , M , M | = T ∀ and M ⊆ M , M and M = M ∩ M .Let M be the following structure. Its universe is just the union of the universes of M , M . For i < P Mi = P M i ∪ P M i . R M , = R M , ∪ R M , and similarly define R M , = R M , ∪ R M , . Let R M , = R M , ∪ R M , ∪ { ( a, b ) : a ∈ P M \ M , b ∈ P M \ M }∪ { ( a, b ) : a ∈ P M \ M , b ∈ P M \ M } . Let us check that (cid:11) holds. Suppose that M | = R , ( b, c ). Then we may assumethat b, c ∈ M (for M it is the same argument). Suppose that M | = R , ( a, b ).Then if a ∈ M then M | = R , ( a, c ). Otherwise a ∈ M and b ∈ M . If c ∈ M as well, then M | = R , ( b, c ) ∧ R , ( a, b ) so M | = R , ( a, c ) and we are done.Otherwise c ∈ M \ M , in which case R M , ( a, c ) holds by choice of R M , . (cid:3) Corollary 5.7. T ∀ has a model completion T which has quantifier elimination. Proposition 5.8. T is NSOP and has SOP .Proof. We start by showing that T is NSOP . Suppose that (cid:104) a i : i < ω (cid:105) is anindiscernible sequence in some model M | = T which witnesses SOP . Let A i be a i as a set. Let M = A , M (cid:48) = A , M = A A , M = A A and M = A A withthe induced structure from M . So all are models of T ∀ . Let M (cid:48) be the amalgamof M , M over M as defined in the proof of Claim 5.6, and similarly let M (cid:48)(cid:48) bethe amalgam of M , M over M (cid:48) . Note that both M (cid:48) and M (cid:48)(cid:48) contain M as asubstructure and that the universe of M (cid:48) is A A A and of M (cid:48)(cid:48) is A A A , butneither are necessarily substructures of M .Now we can amalgamate M (cid:48) and M (cid:48)(cid:48) over M . Moreover, • Any structure N whose universe is A A A A which contains both M (cid:48) , M (cid:48)(cid:48) as substructures and satisfies T ∀ except perhaps (cid:11) , and such that N (cid:22) A A | = T ∀ will be a model of T ∀ (i.e., (cid:11) just follows).To see this, suppose that N | = R , ( b, c ) ∧ R , ( a, b ). We have to show that N | = R , ( a, c ). Note that for every x ∈ N , if x ∈ A i ∩ A j for distinct i, j ∈ { , . . . , } , x ∈ (cid:84) i =1 A i by indiscernibility.If a, b, c all belong to either A A A , A A A or A A then this is clear, soassume this is not the case.Suppose that b, c ∈ A A A , a ∈ A (so a / ∈ A A A ) and b / ∈ A . Then if b ∈ A \ A then M (cid:48)(cid:48) | = ¬ R , ( a, b ) — contradiction, so b ∈ A . Then it must bethat c ∈ A \ A and b ∈ A \ A so M (cid:48) | = ¬ R , ( b, c ) — contradiction.If b, c ∈ A A A , a ∈ A and b ∈ A then c / ∈ A . If c ∈ A \ A then M (cid:48)(cid:48) | = R , ( a, c ) so we are done. Else, c ∈ A \ A , so since b / ∈ A , M (cid:48) | = ¬ R , ( b, c ) —contradiction.Suppose that b ∈ A and c ∈ A . Then a ∈ A A . If a ∈ A \ A then M (cid:48)(cid:48) | = R , ( a, c ) so we are done. Otherwise, a ∈ A \ A , so M (cid:48) | = ¬ R , ( a, b )— contradiction.The case where b ∈ A and c ∈ A is done similarly. By symmetry, this covers all the cases so the bullet is proved.Let σ : A A → A A be a bijection such that σ ( a ) = a and σ ( a ) = a as tuples (hence σ = id). Let N be an amalgam of M (cid:48) and M (cid:48)(cid:48) over M withdomain A A A A . Now define N to be a structure with the same underlying setand the same interpretation of the unary predicates, but with each R i,j interpretedas follows: R Ni,j = (cid:16) R N i,j \ ( A A ) (cid:17) ∪ { ( a, b ) ∈ A A : M | = R i,j ( σ ( a ) , σ ( b )) } . By indiscernibility, if a, b are either both in A or both in A , then ( a, b ) ∈ R Ni,j if and only if ( a, b ) ∈ R Mi,j . Then it is clear that N has underlying set A A A A and extends both M (cid:48) and M (cid:48)(cid:48) , hence it satisfies the conditions in the bullet pointabove. This shows N | = T ∀ , and hence there is some N (cid:48) | = T containing N .But then, if ϕ ( x, y ) is any quantifier-free formula with M | = ϕ ( a , a ), then N (cid:48) | = ϕ ( a , a ) ∧ ϕ ( a , a ) ∧ ϕ ( a , a ) ∧ ϕ ( a , a ). By quantifier elimination, T isNSOP .Next we show that T has SOP . For this we will use the following criterion. Fact 5.9. [14, Claim 2.19] For a theory T , having SOP is equivalent to findingtwo formulas ϕ ( x, y ) , ψ ( x, y ) and a sequence (cid:104) a i , b i : i < ω (cid:105) in some M | = T suchthat • For all i < j , M | = ¬∃ x ( ϕ ( x, a j ) ∧ ψ ( x, a i )). • If i ≤ j then M | = ϕ ( b j , a i ) and if j < i then M | = ψ ( b j , a i ).(The definition in [14] additionally requires that { ϕ ( x ; y ) , ψ ( x ; y ) } is inconsistent,but this added condition is unnecessary: given ϕ and ψ as above, one can replace ϕ by ϕ (cid:48) = ϕ ( x ; y ) ∧ ¬ ψ ( x ; y ) and then ϕ (cid:48) and ψ will witness the above conditions).Let ϕ ( x, y (cid:48) ) = R , ( x, y (cid:48) ) and ψ ( x, y (cid:48)(cid:48) ) = ¬ R , ( x, y (cid:48)(cid:48) ). Let (cid:104) a (cid:48) i , a (cid:48)(cid:48) i , b i : i < ω (cid:105) be a sequence such that R , (cid:0) a (cid:48) i , a (cid:48)(cid:48) j (cid:1) iff i > j , R , ( b j , a (cid:48) i ) whenever i ≤ j and ¬ R , ( b j , a (cid:48)(cid:48) i ) whenever i > j . This sequence exists in some model M | = T aswe can define a model of T ∀ which contains exactly those elements. Now letting a i = ( a (cid:48) i , a (cid:48)(cid:48) i ), the first bullet follows from (cid:11) and the second bullet by the choice of a (cid:48) i , a (cid:48)(cid:48) i and b i . (cid:3) Corollary 5.10. There is a theory T with NSOP having SOP such that Condition5.2 holds with ∆ = { R , ( x, y ) } and θ ϕ from there being R , . Thus T has ∆ -singular compactness by Lemma 5.3.Proof. We only need to show that Condition 5 . M | = T and r is some finite ∆-type. Let A ⊆ M be a finite set of realizations. Now thedefinition of T , we may find some b ∈ M with R , ( b, c ) whenever R , ( x, c ) ∈ r and R , ( a, b ) for all a ∈ A . This suffices. (cid:3) References [1] John T Baldwin, Rami Grossberg, and Saharon Shelah. Transfering saturation, the finitecover property, and stability. The Journal of Symbolic Logic , 64(2):678–684, 1999.[2] Artem Chernikov. 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