Density of compressible types and some consequences

We study compressible types in the context of (local and global) NIP. By extending a result in machine learning theory (the existence of a bound on the recursive teaching dimension), we prove density of compressible types. Using this, we obtain explicit uniform honest definitions for NIP formulas (answering a question of Eshel and the second author), and build compressible models in countable NIP theories.


Introduction
By the Sauer-Shelah lemma, if a formula φ(x; y) is NIP, then the number of φ-types over a finite set A is bounded by a polynomial in the cardinality of A. For a stable formula, this is a consequence of definability of types: one only needs to specify the parameters involved in the definition.In dense linear orders, the reason for this phenomenon is different: for any finite set A and element b, the ≤-type of b over A is implied by its restriction to some subset A 0 of size 2: the information of the full type can be compressed down to this subset of bounded size.A ≤-type over an infinite set cannot in general be compressed down to a finite set, however finite parts of it can be uniformly compressed; following [Sim20], we call such a type compressible (Definition 2.11).We expect NIP formulas to exhibit a combination of those two behaviours.
For NIP theories one manifestation of this philosophy is the result from [Sim20] that an arbitrary type has a generically stable part up to which it is compressible.Distal structures are (NIP) structures in which every type is compressible and hence this decomposition is trivial.For stable theories, compressible types turn out (Lemma 4.8) to be precisely types which are l-isolated, that is, isolated formula by formula.These play a role in Shelah's classification theory; one key property is that in a countable stable theory, an l-atomic model exists over any set [She90, IV.2.18(4),3.1(5),3.2(1)].In this paper, we think of compressibility as an isolation notion and investigate its properties by analogy with the stable case.In order to obtain similar model-construction results, we need two basic properties: density of compressible types and transitivity of compressibility.
Density of compressible types over a set A means that every formula over A extends to a complete compressible type over A. We prove this for countable NIP theories (Corollary 3.21) by first considering the local setting of a single NIP formula φ, and showing that any finite partial φ-type extends to a complete compressible φtype (Corollary 3.9).This is a combinatorial argument based on the proof by Chen, Cheng, and Tang [CCT16] of a bound on the "recursive teaching dimension" of a finite set system in terms of its VC-dimension.The existence of such a bound was used in [EK20] to prove uniform definability of types over finite sets (UDTFS) for an NIP formula in an arbitrary theory.We generalise this result (answering [EK20, Bays was partially supported by DFG EXC 2044-390685587 and ANR-DFG AAPG2019 (Geomod).Kaplan would like to thank the Israel Science Foundation for their support of this research (grant no.1254/18).Simon was partially supported by the NSF (grants no.1665491 and 1848562).
Question 28]) by showing uniformity of honest definitions for NIP formulas, which was previously known only assuming NIP for the whole theory [CS15,Theorem 11].For this, we first show that an arbitrary φ-type p is a rounded average of finitely many compressible types (Theorem 5.17).The rounded average of the compression schemes of these types gives an honest definition for p.
In fact, it turns out that full consistency of p is not required here: for large enough k we get uniform honest definitions for k-consistent families of instances of φ, which we dub k-hypes (Corollary 5.30).Using this, we also obtain in Theorem 5.36 uniform definability of φ-types which are pseudofinite in the sense that their positive and negative parts are pseudofinite (Definition 5.34).
In order to prove transitivity, namely that tp(AB/C) is compressible when tp(A/BC) and tp(B/C) are, we return to the global setting of an NIP theory and use the type-decomposition theorem from [Sim20].We show in Proposition 6.23 that compressibility can be rescoped to an arbitrary subset of the domain: if tp(a/B) is compressible and C ⊆ B, then tp(a/C) is compressible in the language with constants for elements of B. We deduce transitivity in Proposition 6.25.
Finally, we conclude that for countable NIP theories (or even countable theories naming any set of constants) one can construct models which are compressible over arbitrary sets (Propositions 6.29 and 6.30).We give several applications: • Given a definable unary set X whose induced structure is stable, and any model N of the theory of the induced structure, there is a model M of T such that X(M ) = N and moreover, if N ′ ≻ N then there is M ′ ≻ M such that X(M ′ ) = N ′ .This is Corollary 6.33.
• If the theory is not stable, we can extend models without realising any nonalgebraic generically stable type (Corollary 6.39 and Remark 6.40).
• We analyse compressiblity in ACVF, showing that then any model M containing A whose residue field is algebraic over A is compressible over A (Example 6.41).
1.1.Acknowledgements.We thank to Nati Linial and Shay Moran for answering a question that turned out to be precisely about the existence of a bound for the recursive teaching dimension, introducing us to this notion and to [HWLW17].
We also thank Timo Krisam for helpful conversation which led to an improvement in the formulation of Section 6.2, and Eran Alouf for asking questions that led to Theorem 5.36.
Additionally we thank Anand Pillay and Martin Hils for suggesting that we consider the problem that led us to Corollary 6.33.
Furthermore, we thank the anonymous referee for their careful reading of the manuscript and their many useful and precise comments which improved the presentation of the paper.

Preliminaries
2.1.Languages, formulas and types.Our notation is standard.We use L to denote a first order language and φ(x, y) to denote a formula φ with a partition of (perhaps a superset of) its free variables.When x is a (possibly infinite) tuple of variables and A is a set contained in some structure (perhaps in a collection of sorts), we write A x to denote the tuples of the sort of x (and of length |x|) of elements from A; alternatively, one may think of A x as the set of assignments of the variables x to A. When M is a structure and A ⊆ M x , b ∈ M y , we define φ(A, b) = {a ∈ A | M φ(a, b)}.
T will denote a complete theory in L (we do not really need T to be complete, but it is more convenient), and U T will be a monster model (a sufficiently large saturated model 1 ).The word small means "of cardinality < |U|".As usual, we will assume that all models, tuples and sets of parameters are small and are contained in (perhaps a collection of sorts from) U unless stated otherwise.Some results, such as Theorem 5.17, hold for any set, by considering a bigger monster model and applying the result there.
When B ⊆ U, L(B) is the language L augmented with constants for elements from B, and U B is the natural expansion of U to L(B).A partial type in variables x (perhaps infinite, perhaps from different sorts) over B ⊆ U is a set of L(B)-formulas in x consistent with Th(U B ) (i.e., formulas over B).For a partial type π over B and C ⊆ B, we use the notation π| C for the restriction of π to C, namely all formulas φ(x) ∈ L(C) implied by π (i.e., π ⊢ φ(x)).Similarly, if x ′ is a sub-tuple of x, the restriction of π to x ′ is the partial type consisting of all formulas in x ′ implied by π.
A (complete) type over B is a maximal partial type over B. We denote the space of types over B in x by S x (B).It is a compact Hausdorff topological space in the logic topology (a basic open set has the form {p ∈ S x (B) | φ(x) ∈ p}).For a ∈ U x , write tp(a/B) for the type of a over B. S(B) is the union of all types over B.
For an L-formula φ(x, y), an instance of φ over B ⊆ U is a formula φ(x, b) where b ∈ B y , and a (complete) φ-type over B is a maximal partial type consisting of instances and negations of instances of φ over B. We write S φ (B) for the space of φ-types over B in x (in this notation we keep in mind the partition (x, y), and x is the first tuple there).As above, it is a compact Hausdorff topological space in the logic topology.For a ∈ U x , we write tp φ (a/B) ∈ S φ (B) for its φ-type over B. We also use the notation φ 1 = φ and φ 0 = ¬φ.When p(x) ∈ S(B) is a type, we write p ↾ φ ∈ S φ (B) for the complete φ-type over B implied by p.If ∆ is a set of partitioned formulas φ(x, y), we define S ∆ (B) and the restriction p ↾ ∆ ∈ S ∆ (B) similarly.
We will also consider the case where B ⊆ U y and (abusing notation) define S φ (B) similarly -this should never cause a confusion.
If π(x) is a small partial type (over some small set contained in U), we write S π φ (B) for the closed subspace of S x φ (B) consisting of the φ-types which are consistent with π.
Generally we do not limit our discussion to finite tuples of variables (but in the context of φ-types for a formula φ this does not matter).
We write A ⊆ fin B to mean that A is a finite subset of B.

Global and invariant types.
For A ⊆ U, an A-invariant type is a global type, i.e., a type over U, which is invariant under the action of Aut(U/A), the group of automorphisms fixing A pointwise.For a sequence (x i ) i∈I and j ∈ I, we write x <j for (x i ) i<j , and similarly for x ≤j .
Definition 2.1.If q(x) and r(y) are A-invariant global types, then the type (q ⊗ r)(x, y) is defined to be tp(a, b/U) (in a bigger monster model) for any b r and a q| U b (here we understand q to mean its unique A-invariant extension to a bigger model).(This can also be defined without stepping outside of the monster model, see [Sim15a, Chapter 2].) We define q (n) (x <n ) for n < ω by induction: q (1) (x 0 ) = q(x 0 ), and q (ω) (x <ω ) = n<ω q (n) .
1 There are set theoretic issues in assuming that such a model exists, but these are overcome by standard techniques from set theory that ensure the generalised continuum hypothesis from some point on while fixing a fragment of the universe.The reader can just accept this or alternatively assume that U is merely κ-saturated and κ-strongly homogeneous for large enough κ.
For any linear order (X, <), we can define q (X) (x i | i ∈ X) similarly, as the union of q (X0) (x i | i ∈ X 0 ) for every finite X 0 ⊆ X. Fact 2.2.[Sim15a, Chapter 2] Given a global A-invariant type q and a linear order (X, <), q (X) is an A-invariant global type.In addition, it is the type of an indiscernible sequence over U.
For any small set B ⊇ A, q (X) | B is given by tp((a i | i ∈ X)/B) where a i q| Ba<i .This is a Morley sequence of q over B (indexed by X).

VC-dimension and NIP.
Definition 2.3 (VC-dimension).Let X be a set and F ⊆ P(X).The pair (X, F ) is called a set system.We say that A ⊆ X is shattered by F if for every S ⊆ A there is F ∈ F such that F ∩ A = S.A family F is said to be a VC-class on X if there is some n < ω such that no subset of X of size n is shattered by F .In this case the VC-dimension of F , denoted by vc(F ), is the smallest integer n such that no subset of X of size n + 1 is shattered by F .
If no such n exists, we write vc(F ) = ∞.
Definition 2.4.Suppose T is an L-theory and φ(x, y) is a formula.Say φ(x, y) is NIP if for some/every M T , the family The theory T is NIP if all formulas are NIP.
Definition 2.5.Suppose T is an L-theory and φ(x, y) is an NIP formula.Let vc(φ) be the VC-dimension of {φ(M x , a) | a ∈ M y }, where M is any (some) model of T .Note that this definition depends on the partition of variables.Let φ opp be the partitioned formula φ(y, x) (it is the same formula with the partition reversed).Let vc * (φ) = vc(φ opp ) be the dual VC-dimension of φ.
Fact 2.6.[Sim15a, Lemma 6.3] Suppose F is a VC-class on X.Let F * = {{s ∈ F | x ∈ s} | x ∈ X} ⊆ P(F ) be the dual of F .Then F is a VC-class iff F * is, and moreover vc * (F ) := vc(F * ) < 2 vc(F )+1 .Remark 2.7.By Fact 2.6, φ is NIP iff φ opp is NIP, and vc(φ opp ) = vc * (φ) < 2 vc(φ)+1 .By [Sim15a, Lemma 2.9], a Boolean combination of NIP formulas is NIP.In particular, if φ i (x, y i ) is NIP for i < k then so is i<k φ i (x, y i ).We end this subsection by giving an explicit bound on its VC-dimension; see also [DKL84,Theorem 9.2.6].(This will be used only in Section 5.3.).Definition 2.8.Let s ≤k = i≤k s i , and let Lemma 2.10.Let k ∈ N, and let φ 1 (x; y 0 ), . . ., φ n (x; y n−1 ) be partitioned formulas with vc(φ i ) ≤ k.Let θ(x; y 0 , . . ., y n−1 ) = i<n φ i (x, y i ).Then vc(θ) ≤ B vc (n, k).Definition 2.11.A type p(x) ∈ S(A) is compressible if for any formula φ(x, y) there is a formula ψ(x, z) such that for every finite set A 0 ⊆ A, there is some c ∈ A z such that Given a ∈ U x any tuple, we let (A, a) be the structure with universe A and the induced structure coming from a-definable sets.In other words, for every formula φ(x, y), there is a relation R φ (y) interpreted by R φ (c) iff U φ(a, c) for any c ∈ A y .Note that if M ≡ (A, a) then M ∼ = (A ′ , a) for some A ′ ⊆ U, and moreover if M ≻ (A, a) then there is such an A ′ ⊆ U such that M and (A ′ , a) are isomorphic over A. Thus, whenever we have such a structure, we will always assume it has the form (A ′ , a) for some A ′ ⊆ U.
This construction preserves useful information on the type tp(a/A).For example, recall that a type p(x) ∈ S(A) is definable if for every formula φ(x, y), the set {a ∈ A y | φ(x, a) ∈ p} is definable over A. It is easy to see that if tp(a/A) is definable and (A ′ , a ′ ) ≡ (A, a) then tp(a ′ /A ′ ) is also definable (with the same definition scheme).Moreover, we have:

Compactness gives the following equivalent definition of compressibility:
Fact 2.13.The type p = tp(a/A) is compressible iff for any (some) |A| + -saturated elementary extension (A ′ , a) ≻ (A, a) and any formula φ(x, y), there is some formula ψ(x, z) and d ∈ (A ′ ) z such that ψ(a, d) holds and ψ(x, d) ⊢ (p ↾ φ)| A .
In fact, this was the original definition of compressibility in [Sim20, Definition 3.1].
We give another useful characterisation of compressible types.Recall that two types p(x), q(y) over A are weakly orthogonal if p ∪ q implies a complete type in x, y over A.
A type q is finitely satisfiable in some set A if every formula from q is realised in A. We write S x A-fs (B) ⊆ S x (B) for the subspace consisting of those types in x which are finitely satisfiable in A. Recall that such types can be extended to global types in S x A-fs (U) (using ultrafilters).Note that S x A-fs (B) is a closed (and hence compact) subspace of S x (B).As usual, omitting the x means taking all types (allowing infinite (small) tuples).
(2) For all q(y) ∈ S A-fs (A ′ ), tp(a/A ′ ) (as a type in x) and q(y) are weakly orthogonal.
(3) For all q(y) ∈ S A-fs (A ′ ), tp(a/A ′ ) (as a type in x) and q(y) imply a complete type in xy over ∅.
The other direction follows from Fact 2.14(3⇒1), since by saturation we can assume d ⊆ A ′ .

Density of (local) compressibility
Here we will prove that (local) compressible types are dense.In Section 3.1 we prove an abstract version of this dealing with set systems of finite VC-dimension (generalising [CCT16, Lemma 4] to infinite sets).Then in Section 3.2 we deduce that locally compressible types are dense for NIP formulas, and in Section 3.3 we deduce that compressible types are dense in countable NIP theories.
3.1.Compressibility for set systems of finite VC-dimension.Let A be a (possibly infinite) set.As usual, 2 A is the (Hausdorff compact) space of functions A → 2 = {0, 1} equipped with the product topology.Any C ⊆ 2 A naturally induces a set system on A (those sets whose characteristic functions are in C) and as such has a VC-dimension vc(C).For C ⊆ 2 A and B ⊆ A, let Remark 3.2.This terminology is originally inspired by, but does not precisely agree with, the terminology around compression schemes in the statistical learning literature.
Remark 3.3.Suppose (P, ≤) is a directed partial order, and c : P → r < ω is some colouring.Then there is some subset X ⊆ P which is monochromatic (X ⊆ c −1 (i) for some i < r) and cofinal (for all p ∈ P there is some q ∈ X such that q ≥ p).
Indeed, if not, then for every i < r there is some p i ∈ P such that c(q) = i for all q ≥ p i .Let p ≥ p i for all i < r.Then c(p) = i for all i < r, contradiction.
The proof of the following theorem is an adaptation to the case of infinite A of the proof of [CCT16, Lemma 4], which proves it for finite A with the same bound on k.Proof.We may assume |A| ≥ k 0 , as otherwise the result is immediate (take D = A and any c ′ ∈ C).
Equip S with the partial order of inclusion, i.e., C| We claim that there is e ∈ 2 B such that c ′ ∪ e ∈ S, contradicting maximality of c ′ .Indeed, let A 0 ⊆ fin D ∪ B and let In this way we obtain a 2 B -colouring of the partial order of finite subsets of D ∪ B, where each finite A 0 ⊆ D ∪ B is coloured with an e ∈ 2 B such that (1) holds for some Remark 3.6.For finite A, the exponential dependency of k comp (d) on d obtained in [CCT16] was improved to a quadratic dependency in [HWLW17].Conjecturally it is even linear (see the introduction to [HWLW17]).The proof of this quadratic bound does not adapt so readily to the infinite case, and it would be interesting to find the best bound, and in particular to see whether Theorem 3.4 holds with a quadratic bound.

Density of compressible local types.
In the following definition, we use the notation p ⊢ π q for p ∪ π ⊢ q, where p, π are small partial types and q a finite partial type (this is compatible with the notation in Definition 3.1 when p, q are complete φ-types, and C is the set of φ-types consistent with π).As usual, we work in a complete L-theory T .Definition 3.7.Fix a formula φ(x, y), a parameter set A ⊆ U y and a small partial type π(x).Recall the notation S π φ (A) from Section 2.1.
A) considered as a (closed) subspace of 2 A as in Definition 3.1: for any finite it is a countable union (going over all k) of countable intersections (going over all finite subsets of A) of countable unions (going over all subsets of A of size ≤ k) of clopen sets (the implication).
Corollary 3.9.Let φ(x, y) be a formula, d ∈ N, A ⊆ U y and π(x) a small partial type.Suppose that φ(x, y) is NIP and that vc * (φ Corollary 3.10.The following are equivalent for a formula φ(x, y) and a partial type π(x).
(2) There exists k < ω such that for any set This gives a new characterisation of NIP types.Definition 3.11.We say that a partial type π(x) has IP if there is a formula φ(x, y) ∈ L which has IP as witnessed by realisations of π, i.e., if vc({φ(π(U), a) : a ∈ U y }) = ∞.A formula or a partial type is NIP if it does not have IP.

By compactness we have that:
Remark 3.12.A partial type π(x) is NIP iff for every formula φ(x, y) there is a formula ψ(x) implied by π such that ψ(x) ∧ φ(x, y) is NIP (as a formula over U).
By Corollary 3.10 and Remark 3.12 we have: Corollary 3.13.A partial type π(x) is NIP iff for every formula φ(x, y) and A ⊆ U y there is k < ω such that S π φ↓k (A) = ∅.3.3.Density of compressible types in countable NIP theories.Now we turn from local types to types.Definition 3.14.Let π(x) ⊆ π ′ (x) be partial types over a parameter set A ⊆ U (perhaps contained in a collection of sorts).
(1) p is compressible if for each formula φ(x, y), the restriction p ↾ φ ∈ S φ (A) of p to a φ-type is compressible within p with respect to A.
(2) p is strongly compressible if for each formula φ there exists a finite set of formulas ∆ ∋ φ such that p ↾ ∆ is t-compressible with respect to A.
Remark 3.15.Note that the definition above of a compressible type is the same as Definition 2.11.
Remark 3.16.The reason we say "with respect to A" in the definition is because a partial type over A is also a partial type over any set containing A. In the future we will usually omit this since A will be clear from the context.
Remark 3.17.As we said in Section 2.1, we do not restrict ourselves to finitary types.Note that p ∈ S x (A) is compressible iff all of its restrictions to finite tuples of variables are compressible.
Remark 3.18.The relations between these definitions and the definitions for φtypes in Definition 3.7 are slightly subtle.In particular, for A ⊆ U y and a φ-type p ∈ S φ (A), the condition that p is ⋆-compressible (i.e., k-compressible for some k) is strictly stronger than the condition that p is t-compressible.For example, in Th(N; <), the non-realised (x = y)-type in S x=y (N) is t-compressed by x > z, but is not k-compressible for any k < ω.
Remark 3.19.Note that for a model M and p ∈ S(M ), p is (strongly) compressible iff its unique extension p eq to M eq is (strongly) compressible (by translating formulas in L eq to formulas in L, see [Pil96, Lemma 1.1.4]).
Lemma 3.20.Let π(x) be a t-compressible partial type over a set A ⊆ U y , and let φ(x, y) be an NIP formula.Then there exists p φ ∈ S φ (A) such that π ∪ p φ is consistent and t-compressible.Moreover, there is a formula ξ(x, w) which is a Boolean combination (depending only on vc * (φ)) of instances of φ and equality such that if ζ(x, w ′ ) t-compresses π (i.e., compresses π within itself ) then ζ(x, w ′ ) ∧ ξ(x, w) t-compresses π ∪ p φ .
Proof.We may assume |A| > 1, as otherwise the result is clear.
By Corollary 3.9(i), there is p φ ∈ S π φ↓k (A) for some k < ω depending only on vc * (φ).By a coding of finitely many formulas as one as in the proofs of e.g.[She90, Theorem II.2.12(1)] and [Gui12, Lemma 2.5], we obtain a formula ξ(x, w) such that for any finite A 0 ⊆ A, there is a ∈ A w such that p φ (x) ⊢ ξ(x, a) and π ∪ {ξ(x, a)} ⊢ p φ | A0 .Explicitly, we may take ξ(x, w) with w = (w i j ) i<3,j<k to be j<k (φ(x, w 0 j ) ↔ w 1 j = w 2 j ).Then, for any finite A 0 , there is some Finally, let a = (a i j ) i<3,j<k .Now assume that ζ(x, w ′ ) is as in the lemma and fix some finite set A 0 ⊆ A. Let a ∈ A w be as above.By compactness there is a finite , and so (by the assumption on ζ) there is Corollary 3.21.(T countable NIP) Suppose A ⊆ U is a set of parameters and x is a countable tuple of variables.Then, compressible types are dense in S x (A): If θ(x) is a consistent formula over A, then there exists a compressible type p(x) ∈ S(A) with p(x) ⊢ θ(x).
More generally, if π(x) is a t-compressible partial type over A, then there exists a strongly compressible p ∈ S(A) with π ⊆ p.
Proof.Clearly it is enough to prove the "more generally" part, so assume π is t-compressible and ζ compresses π within π.
Enumerate the formulas φ(x, y) as (φ i (x, y i )) i<ω (where the y i 's are finite), with Recursively applying Lemma 3.20, let p φi ∈ S φi (A) be such that π i+1 := π i ∪ p φi is t-compressible, and moreover is compressed by a Boolean combination of formulas from ∆ i+1 .Then each π i ↾ ∆ i is t-compressible, and so p := i<ω π i is strongly compressible.
For an example showing the necessity of the countability assumption, see Remark 4.11 below.
Remark 3.22.It follows from Corollary 3.10 that Lemma 3.20 characterises φ being NIP (letting π be the empty type).However, Corollary 3.21 does not characterise NIP for countable theories.An easy example is Th(N, +, •), and in fact any theory with IP in which dcl(A) is a model for any set A (given a consistent formula θ(x) over a set A, let c θ be in dcl(A), then tp(c/A) is compressible and even isolated).
Question 3.23.We could consider an apparently weaker notion of compressibility of a type: say p ∈ S(A) is weakly compressible if for any formula φ(x, y) there is some formula ζ(x, z) such that for any finite A 0 ⊆ A there is some weak compressibility is equivalent to compressibility, but for general sets it is less clear.In Example 6.26 below we will see that if T is the theory of atomless Boolean algebras, this can fail.Is it true that if T is NIP then p is weakly compressible iff p is compressible?

Compressibility and stability
Here we discuss compressibility in the context of stability, in both the local and global senses, and point out that compressibility is equivalent to l-isolation (see Definition 4.6) in these contexts.The main results are: • For stable formulas, k-compressibility is equivalent to k-isolation (Lemma 4.3).
• For stable types, compressibility is equivalent to l-isolation (Lemma 4.8), and in particular when T is stable these two notions are the same.• For generically stable types, compressibility is equivalent to l-isolation (Proposition 4.14).4.1.Stable formulas.Recall that a formula φ(x, y) is stable if it does not have the order property: there are no (a i , b i ) i<ω such that φ(a i , b j ) holds iff i < j, and φ has the strict order property (SOP) if there is a sequence (b i ) i<ω such that (φ(U x , b i )) i<ω forms a strictly decreasing sequence of definable sets (with respect to containment).A theory T is stable if no formula has the order property.Clearly if φ is stable, it is NIP.
The following says in particular that under stability, k-compressibility and kisolation are the same.Proof.(i) implies (iii) as a Boolean combination of stable formulas is stable (see e.g., [Pil96, Lemma 2.1]) and the strict order property implies the order property.
Inductively we find b i ∈ B k and ǫ i ∈ {0, 1} k for i < ω such that we have ) But then some ǫ occurs infinitely often, and then θ ǫ (x, y) has SOP.(1) π is stable.
(2) For every formula φ(x, y) there is a formula ψ(x) implied by π such that φ(x, y) ∧ ψ(x) is stable (as a formula over U).
Under stability, the analogue of compressibility of a type is l-isolation.
Clearly, an l-isolated type is compressible.By considering the formula x = y, we easily obtain: Remark 4.7.Any l-isolated type over a model is realised.
The following is analogous to (but not actually comparable with) Lemma 4.3.
Lemma 4.8.(i) Suppose p ∈ S(A) is compressible but not l-isolated.Then: (a) There are tuples a i , b i in A which witness the order property for some L-formula.(b) p is not stable.In particular, if T is stable then any compressible type is l-isolated.(ii) (T countable NIP) T is stable iff any compressible type is l-isolated, iff there is some ω-saturated model M such that every strongly compressible type over M is l-isolated.
Proof.For both (i.a) and (i.b), suppose φ(x, y) witnesses that p is not l-isolated and ζ(x, z) compresses p ↾ φ. j≤i θ(x, b j ) does not isolate p ↾ φ, there are some a i , c i such that j≤i θ(a i , b j ), φ(x, c i ) ǫ ∈ p for some ǫ < 2, and ¬φ(a i , c i ) ǫ holds.
(ii) The implications from left to right follow by (i) and trivially, respectively.For the other direction, assume that T is not stable.By [Sim15a, Theorem 2.67], T has the SOP.So say < is an ∅-definable (strict) preorder on an ∅-definable set D with infinite chains.
Let M be an ω-saturated model.So M contains an infinite chain C which we may assume is maximal.Since C is infinite, we can write C = C 1 + C 2 where either C 1 has no last element or C 2 has no first element (one of them may be empty).
Let π(x) be the unique type in nonempty, then z < x < w compresses π within π and if C 1 is empty then x < z compresses π within π).Hence by Corollary 3.21, π has a strongly compressible completion q ∈ S(M ).By maximality of C, π is not realised in M , so neither is q.So by Remark 4.7, q is not l-isolated.
(Note that we could have worked with a partial order instead of a preorder by passing to eq and using Remark 3.19.)Remark 4.9.From the proof of Lemma 4.8(ii) it follows that if M T contains an infinite chain in an M -definable preorder D then there is c ∈ D such that tp(c/M ) is compressible but not l-isolated.
Remark 4.11.The following example demonstrates the necessity of the countability assumption on T in Corollary 3.21 even for stable theories.
Let κ be a cardinal, and consider κ colourings on a set X, with each colouring using the same colours, such that no point gets the same colour according to two different colourings, but apart from this restriction all possibilities are realised.We can formalise this in the language with a sort X, a sort C for the colours, and for each i ∈ κ a function f i : X → C giving the colour of an element according to the i-th colouring.The theory is axiomatised by saying there are infinitely many colours and, for each finite set {i 1 , . . ., i n } ⊆ κ enumerated without repetitions and each m ≥ 0, an axiom This axiomatises a complete consistent theory T with quantifier elimination in the given language.Indeed, restricting to any finite sublanguage L 0 containing X, C and finitely many function symbols f i , T ↾ L 0 is the Fraïssé limit of the class of finite structures (X 0 , C 0 ) where for every f i , f j ∈ L 0 and all x ∈ X 0 , if We claim that the formula x ∈ X has no compressible (equivalently, by Lemma 4.8(i), l-isolated) completion p ∈ S x (C 0 ).Indeed, it is easy to see that p would have to include for each i a formula f i (x) = c i for some c i ∈ C 0 , but then c i = c j for i = j, contradicting κ > |C 0 |.
Other (hints for) examples are given in [She90, Exercise IV.2.13] where Shelah also gives a superstable3 counterexample, which we will describe briefly.Let L = {E ν , P s | ν ∈ ω ω , s ∈ ω <ω } where the P s 's are unary predicates and the E ν 's are binary relation symbols.Let M be the L-structure whose universe is ω ω × ω where Essentially, each class is infinite and two branches in the tree ω ω are ν-equivalent if they divert from ν at the same point and in the same direction (starting the same cone), and ν is ν-equivalent only to itself.Let T = Th(M ).It is not too hard to see that T has quantifier elimination.We leave it as exercise to check that for any set A, S 1 (A) ≤ |A|+2 ℵ0 , and thus T is superstable.
Finally, working in U eq and letting A = {(η, n)/E ν | η = ν, n < ω}, there is no l-isolated type p ∈ S 1 (A) (in the home sort).Indeed, if p(x) is such a type, then for every ν ∈ ω ω there is some η = ν such that x/E ν = (η, 0)/E ν is in p.It follows that for some ν ∈ ω ω , p must contain {P s (x) | s ⊳ ν}.But then p ⊢ x/E ν = (η, 0)/E ν for all η = ν, contradiction.4.3.Generically stable types.Generically stable types are invariant types that exhibit stability-like behavior "generically" i.e., when considering their Morley sequences.This notion was first studied in the NIP context by Shelah [She04] (under the name "stable types") and then by Hrushovski and Pillay [HP11] and independently Usvyatsov [Usv09].See also [Sim15a, Section 2.2.2].It was defined in general in [PT11] by Pillay and Tanović.See also [CG20] for more on generic stability outside of the NIP context.Definition 4.12.We say that a global type p is generically stable over A if it is A-invariant and for every ordinal α, every φ(x) with parameters in U and every Morley sequence (a i ) i<α of p over A, the set {i < α | U φ(a i )} is finite or cofinite.(3) [CG20, Proposition 3.2] p = lim(a i ) i<ω for any Morley sequence (a i ) i<ω of p over A, i.e., p is the limit type of any of its Morley sequences over A: θ(x) ∈ p iff θ(a i ) holds for all but finitely many i < ω.(In [CG20, Proposition 3.2] it is stated over models, but it is also true over sets and follows directly from the definition; we leave this to the reader.) Moreover, by [Sim15a, Theorem 2.29] if T is NIP then each one of these conclusions is equivalent to generic stability for an A-invariant type.
Proposition 4.14.Let p ∈ S(U) be generically stable over A ⊆ U, and suppose p| A is compressible.Then p| A is l-isolated.
For the proof we will need the following observation.Recall the notations from Section 2.4.
Remark 4.15.Suppose tp(a/A) is l-isolated, and (A, a) Thus, the same is true in (A ′ , a ′ ), which suffices.
Proof of Proposition 4.14.Suppose p ∈ S(U) is generically stable over A and p| A is compressible.Let a p| A , and let M be a model containing Aa.Let (M ′ , A ′ , a) ≻ (M, A, a) be an |M | + -saturated extension (in a language with a predicate P for A and constant symbols a), and let (M ′′ , A ′′ , a) ≻ (M ′ , A ′ , a) be an |M ′ | + -saturated extension (with M ′′ ⊆ U).Since p is definable over A by Fact 4.13(1), it follows that p| A ′′ = tp(a/A ′′ ).
Let φ(x, y) be any formula.Note that (A ′′ , a) ≻ (A ′ , a) ≻ (A, a) and that (A ′′ , a) is |A ′ | + -saturated, so by Facts 2.12 and 2.13, there is some d ∈ (A ′′ ) z and some formula By Fact 4.13(3) and compactness there is some N < ω such that for every Morley sequence (a i ) i<N of p over A, ζ(a i , d) holds for some i < N , and hence Let (a i ) i<n be a Morley sequence of p over A of maximal length such that a i (p ↾ φ)| A ′ for all i < n.For i < n, let c i ∈ (A ′ ) y and ǫ i < 2 be such that φ(a i , c i ) ǫi holds but ¬φ(x, c i ) ǫi ∈ p. Then the following set of formulas over M ′ is inconsistent: where θ(x) = ∀y ∈ P (φ(a, y) ↔ φ(x, y)).By compactness (and saturation of M ′ ), there is some formula ψ(x 0 , . . ., Since φ was arbitrary, this means that p| A ′ is l-isolated, and hence by Remark 4.15, we are done.
It is convenient to use the following definition.

Rounded averages of compressible types and applications
Let Maj be the majority rule Boolean operator, i.e., for truth values P 0 , . . ., P n−1 , let We just write Maj i if n is clear.
The rounded average of p 0 (x), . . ., p n−1 (x) ∈ S φ (B) is the following (possibly inconsistent) collection of formulas The main result of this section is: Theorem 5.2.Let φ(x, y) be an NIP formula and suppose α ∈ [1/2, 1).Then there exist n and k depending only on vc(φ) and α such that for A ⊆ U y , any p ∈ S φ opp (A) is the α-rounded average of n types in S φ opp ↓k (A).
(We give a more precise and general statement in Theorem 5.17, allowing a partial type π(x).) We give some applications: • Uniformity of honest definitions for NIP formulas, see Definition 5.22.This is Corollary 5.23.• Uniform definability of pseudofinite types, see Theorem 5.36.5.1.Superdensity.In this section we isolate a sufficient condition for proving Theorem 5.2, which uses the (p, q)-theorem (see Fact 5.4).We then apply it to retrieve UDTFS in Corollary 5.14, as a prelude to the proof of the uniformity of honest definitions in Corollary 5.23.Definition 5.3.Suppose q ≤ p < ω.A set system (X, F ) has the (p, q)-property if for any S ⊆ F such that |S| ≥ p, there exists S 0 ⊆ S of size |S 0 | ≥ q such that S 0 = ∅.
Fact 5.4.[Mat04] (The (p, q)-theorem) There exists a function N pq : N 2 → N such that for any q ≤ p < ω, if (X, F ) is a finite set system with the (p, q)-property such that every s ∈ F is nonempty and vc * (F ) < q, then there is We isolate from the proof of [Sim15a, Corollary 6.11] the following immediate generalisation of the (p, q)-theorem to infinite set systems.
Lemma 5.5.Let φ(x, y) be NIP.Let p ≥ q > vc * (φ) be integers, and let N = N pq (p, q).Let A ⊆ U x and B ⊆ U y .Suppose that φ(A, b) = ∅ for every b ∈ B, and that for every B 0 ⊆ B with |B 0 | = p there exists B 1 ⊆ B 0 with |B 1 | = q such that for some a ∈ A we have b∈B1 φ(a, b).
Proof.By the definition of finite satisfiability, it suffices to see this in the case that B is finite; but this case is a direct consequence of the (p, q)-theorem (Fact 5.4).Suppose φ(x, y) is a formula, A ⊆ U x and N ∈ N.For variables x = (x i | i < N ) of the same sort of x, we denote by S x φ,A-fs (U y ) the space of ∆-types in x over U y which are finitely satisfiable in A, where ∆ = {φ(x i , y) | i < N }.If p(y) ∈ S φ opp (A) then for any q ∈ S x φ,A-fs (U y ), the product q(x) ⊗ p(y) is the partial type q(x) ∪ p(y) ∪ {φ(x i , y) ǫi | i < N } where ǫ i < 2 is the truth value of φ(a i , b) for some (any) b p and (a i | i < N ) q| b .Note that this is well-defined.
For b ∈ U y , N ∈ N and S ⊆ S φ opp (A), we consider the following condition: For every q(x) ∈ S x φ,A-fs (U y ) where x = (x i | i < N ), there is p ∈ S such that Definition 5.6.Suppose φ(x, y) is a formula and A ⊆ U x .A set S ⊆ S φ opp (A) is superdense in S φ opp (A) if ( †) b,N,S holds for every b ∈ U y and N ∈ N.
Remark 5.7.By considering realised types in A, it follows that any superdense set S ⊆ S φ opp (A) is also dense.
Then r := tp φ opp (b/A) is the α-rounded average of n elements of S.
Note that vc(φ ′ ) = vc(φ) ≤ d; indeed, φ ′ and φ shatter the same subsets of U x .Now assume the conclusion fails.In particular, r / ∈ S (else it is the rounded average of n copies of itself).Let B ⊆ U y be a set of realisations of the types in S.
By assumption, for every B 0 ⊆ B with |B 0 | = n, there exists a ∈ A and ) such that ¬φ ′ (a, c) holds for all c ∈ B (the last inequality follows from Remark 2.7).
Extend π to q(x) ∈ S x φ,A-fs (U y ) (formally, first extend π to a global type finitely satisfiable in A, and then restrict to a global {φ(x i , y) | i < N }-type q.Then note that q implies π).Then for all p ∈ S, we have q(x) ⊗ p(y) ⊢ i<N ¬φ ′ (x i , y).However, ( †) b,N,S implies that for some p ∈ S, q(x) ⊗ p(y) ⊢ i<N φ ′ (x i , y), contradiction.
Remark 5.10.From the proof of Lemma 5.8, we get something slightly stronger (under the same assumptions): either r ∈ S, or r is an α-rounded average of distinct types in S.
Remark 5.11.Lemma 5.8 admits a partial converse, for an arbitrary formula φ(x, y) and any n and any N : letting x = (x i | i < N ), if tp φ (b/A) is the rounded average of n elements from S ⊆ S y φ opp (A), then for every q(x) ∈ S x φ,A-fs (U y ) there is p(y) ∈ S such that q(x) ⊗ p(y) Indeed, suppose tp φ opp (b/A) = rAvg(tp(c j /A) | i < n) where tp(c j /A) ∈ S for j < n and q(x) ∈ S x φ,A-fs (U y ).If the conclusion fails, we get that for each j < n q ⊢ ¬ Maj i<N (φ(x i , c j ) ↔ φ(x i , b)).
By finite satisfiability, there is (a 0 , . . ., a N −1 ) ∈ A N satisfying this for every j < n.Hence |{(i, j) ∈ N × n | φ(a i , c j ) ↔ φ(a i , b)}| ≤ 1 2 nN .Then by the pigeonhole principle, for some i < N we have contradicting tp φ opp (b/A) being the rounded average of the tp(c i /A).
In Section 5.3 we will prove that S φ opp ↓⋆ (A) is superdense in S φ opp (A) and even in a uniform way, as in the proof of Corollary 3.9(iii), which will imply Theorem 5.2 by Lemma 5.8.In the finite case we can already conclude the following, basically because superdensity is the same as density when A is finite.
Corollary 5.12.Fix α ∈ [1/2, 1) and some d ∈ N. Then there are k, n ∈ N (depending only on d, α) such that if φ(x, y) is a formula with vc(φ) ≤ d, then for any finite A ⊆ U x , every r(y) ∈ S φ opp (A) is the α-rounded average of n types in S φ opp ↓k (A).
Proof.Let n, N be as in Lemma 5.8, and let k = k comp (d) + N .Fix some b ∈ U y .By Lemma 5.8, to show the conclusion for tp φ opp (b/A), it is enough to show ( †) b,N,S with S = S φ opp ↓k (A).But every q(x) ∈ S x φ,A-fs (U y ) is realised in A (since A is finite), so it boils down to showing that for every ā ∈ A N , there is some p(y) ∈ S φ opp ↓k (A) such that p(y) ⊢ i<N (φ(a i , y) ↔ φ(a i , b)).This follows from the choice of k and Corollary 3.9(ii) applied to φ opp .
As a corollary we retrieve UDTFS.First recall the definition.Definition 5.13 (UDTFS).We say that φ(x, y) has uniform definability of types over finite sets (UDTFS) if there exists a formula ψ(x, z) such that for every finite set A ⊆ U x with |A| ≥ 2 the following holds: for every b ∈ U y there exist c ∈ A z such that ψ(A, c) = φ(A, b).
Every formula with UDTFS is easily NIP (see e.g., the proof of Theorem 14 in [EK20]).The proof of UDTFS for NIP formulas in [EK20] roughly goes by showing Corollary 5.12 with α = 1/2, and deducing UDTFS from that (this is not stated explicitly in this language, see the proof of Theorem 14, (1) implies (2), (3) there).We omit the details here since we will prove uniformity of honest definitions in Corollary 5.23 below.In Theorem 5.36 we will extend UDTFS to pseudofinite types (see Definition 5.34).
We do point out that the proof here is, at least conceptually, simpler than the proof in [EK20]: both proofs use the finite version of Corollary 3.21, but here the only other ingredient is the (p, q)-theorem, while there both the VC-theorem and von Neumann's minimax theorem are used.

5.2.
A variant of Ramsey's theorem for finite subsets.Here we will prove a variant of Ramsey's theorem for finite subsets of a cardinal.This result generalises Remark 3.3 for (P fin (κ), ⊆) in the same way that Ramsey's theorem generalises the pigeonhole principle.It will be used in the proof of Theorem 5.2.
For κ a cardinal, let P fin (κ) be the set of finite subsets of κ, partially ordered by inclusion.
Say f : P fin (κ) → P fin (κ) is strictly increasing if s t ⇒ f (s) f (t) for all s, t ∈ P fin (κ), and say f is cofinal if for all s ∈ P fin (κ) there is t ∈ P fin (κ) such that f (t) ⊇ s.Note that the image of an n-chain under a strictly increasing map is an n-chain.
Proposition 5.15.Let κ be an infinite cardinal, 0 < n ∈ N and let c : P fin (κ) n < → r < ω be a finite colouring of the n-chains.Then there is a strictly increasing cofinal map f : P fin (κ) → P fin (κ) such that the image of all n-chains f (P fin (κ) n < ) is monochromatic, i.e., |(c • f )(P fin (κ) n < )| = 1.Proof.Denote by M the structure (P fin (κ), ⊆, (P k ) k<r ) where As π is finitely satisfiable in M , there is q ∈ S M-fs (N ) extending π.Let (a n−1 , . . ., a 0 ) q (n) | M and let ā = (a 0 , . . ., a n−1 ).Note that ā ∈ N n < (because for all a q| M and all b ∈ M , b a), and hence for some k < r, ā ∈ P N k .We claim that this colour k works.
The construction is possible because q is finitely satisfiable in M and since there only finitely many conditions to fulfill for each s ∈ P fin (κ) (and because of the choice of ā): given s ∈ S m and a chain as in (i), by induction we have q(x) ⊢ P k (f (s 0 ), . . ., f (s i−2 ), x, a i , . . ., a n−1 ); since there are only finitely many such chains to consider and also q(x) ⊢ x f (t) ∪ s for any t s, we can find f (s) by finite satisfiability.Now the i = n case of (i) implies that the image under f of any n-chain has colour k.Meanwhile (ii) implies that f is cofinal, and that f is strictly increasing.5.3.The proofs of superdensity and of Theorem 5.2.In this section we will prove Theorem 5.2, by proving superdensity of compressible types in a uniform way.Let φ(x, y) be a formula, d ∈ N, and assume that vc(φ Then ( †) b,n,S holds.
Proof.Let x = (x i | i < n) and let q(x) ∈ S x φ,A-fs (U y ).We first reduce to the case (*) n = 1 and q ⊢ φ(x, b).
For i < n, let ǫ i < 2 be such that q ⊢ φ(x i , b) ǫi .Let φ ′ (x, y) = i<n φ(x i , y) ǫi , and consider the global φ ′ -type q ′ (x) ∈ S x φ ′ (U y ) implied by q, which is finitely satisfiable in A.
So p ′ implies a complete type p ∈ S π φ opp (A).Let A 0 ⊆ A be a finite subset.Then for some finite It remains to prove the proposition assuming (*), so assume that n = 1 and q(x) ⊢ φ(x, b).
If A is finite then q is realised in A, and we conclude (as in Corollary 5.12) by Corollary 3.9(ii) applied to φ opp (note that k comp (d) + 2d + 2 ≥ k comp (d) + 1 which would be enough in this case).So suppose κ := |A| ≥ ℵ 0 .
Let F fin (κ) be the filter on P fin (κ) generated by {X s | s ∈ P fin (κ)} where By [Sim15b, Lemma 2.9], q is the limit of a sequence of types realised in A: where a s ∈ A. (Note that in [Sim15b, Lemma 2.9], the type is over a model, but the same statement, with the same proof, works also over a set.)This means that for any c ∈ U y , φ(x, c) ∈ q iff {s | φ(a s , c)} ∈ F fin (κ) iff for some s ∈ P fin (κ), φ(a t , c) holds for all t ⊇ s. (Since q is complete, in fact the left-to-right implication suffices: q = lim s→F fin (κ) (tp φ (a s /U y )) iff we have that {s | φ(a s , c)} ∈ F fin (κ) whenever φ(x, c) ∈ q.) Since q ⊢ φ(x, b), we may assume that φ(a s , b) for all s (indeed, if s 0 is such that φ(a s , b) for all s ⊇ s 0 , then we can ensure this by replacing a s with a s∪s0 ).
Let m = 2 vc(φ) + 3. Let c π m be the 2 2 m -colouring of m-chains in P fin (κ) indicating which Boolean combinations of the corresponding m instances of φ are consistent with π, i.e., c π m (s 0 , . . ., s m−1 ) = {(ǫ i ) i<m ∈ 2 m | i<m φ(a si , y) ǫi ⊢ π ⊥}.By Proposition 5.15, we may assume that c π m is constant; indeed, if f is as in Proposition 5.15, we may replace a s with a f (s) , and then q will still be the limit since f is cofinal.We will refer to the property that c π m is constant as c π m -homogeneity.Identify i ∈ N with {0, . . ., i−1} ∈ P fin (κ).Take a φ opp -type p 0 (y) ∈ S π φ opp ((a i ) i<m ) which first strictly alternates maximally and then is constantly true; i.e., p 0 (y) ⊢ φ(a i , y) ↔ ¬φ(a i+1 , y) for i < l and p 0 (y) ⊢ φ(a i , y) for i ∈ [l, m), and l < m is maximal such that such a type exists.Note that tp(b/(a i ) i<m ) is of this form with l = 0, so some such p 0 exists (here we use the fact that b π).By c π m -homogeneity (in fact c π vc(φ)+1 -homogeneity is enough) and the usual argument for bounding alternation number (see [Sim15a,Lemma 2 and let A ′ = {a 0 , . . ., a l } ∪ {a s | l ⊆ s ∈ P fin (κ)} be the domain of p 1 .
Proof.Suppose A ′ 0 ⊆ fin A ′ .Let s 1 ∈ P fin (κ) strictly contain all sets of the form l ∪ s ∈ P fin (κ) such that a s ∈ A ′ 0 .Let s 1 . . .s m−l−2 be an m − l − 2-chain starting with s 1 .Let Then by c π m -homogeneity, p ′ 0 is consistent with π since p 0 is.Suppose l s 0 s 1 .We claim that p ′ 0 ⊢ π φ(a s0 , y).Otherwise, by c π m -homogeneity, is consistent with π.Since l + 2 < m, this contradicts the maximality of l.
We can now deduce Theorem 5.2.
Theorem 5.17.Let d ∈ N and α ∈ (1/2, 1].Then there exist n and k depending only on d, α such that the following holds.If φ(x, y) is a formula such that vc(φ) ≤ d, then for any A ⊆ U x and a (small) partial type π(y), any p ∈ S π φ opp (A) is the α-rounded average of n types in S π φ opp ↓k (A).Namely, we may take n : Proof.By Lemma 5.8 it is enough to show ( †) b,N,S where N = N pq (n, 2 d+1 ) and S = S π φ opp ↓k sd (N,d) (A), which follows by Proposition 5.16.
Remark 5.18.By Remark 5.10, in the context of Theorem 5.17, if p is not kcompressible then it is an α-rounded average of n distinct types in S π φ opp ↓k (A).We give some immediate corollaries.
We can also improve Lemma 4.3 to add another equivalence: Corollary 5.20.The following are equivalent for an NIP formula φ(x, y): (i) φ is stable.
(ii) For any model M T and any k ∈ N, any p ∈ S φ↓k (M y ) is isolated.
Proof.(i) implies (ii) follows from Lemma 4.3.Remark 5.21.Suppose φ(x, y) is stable.Then we can replace k sd in Proposition 5.16 by a linear (as opposed to exponential, see Remark 2.9) bound in terms of n with a simpler proof.Let x = (x i | i < n), q(x) ∈ S x φ,A-fs (U y ), π(y) and b ∈ π(U) be as there.For i < n, let q i (x i ) = q ↾ {φ(x i , y)}.As q i is finitely satisfiable in A, by [TZ12, Exercise 8.3.6],q i is definable by a Boolean combination of instances of φ opp over A (the exercise assumes that T is stable but this is not necessary).The size of this Boolean combination depends only on φ (really only on the size of a maximal witness for the order property).Hence there is l depending only on φ, and a i,j ∈ A, ǫ i,j < 2 for j < l, such that for some formula θ i (y) of the form j<l φ(a i,j , y) ǫi,j , θ i (b) holds and if b ′ θ i (y) then φ(x i , b) ∈ q iff φ(x i , b ′ ) ∈ q.Let θ(y) = i<n θ i .Note that θ(b) holds, so that θ is consistent with π.
Definition 5.22.[Sim15a, Definition 3.16 and Remark 3.14] Suppose φ(x, y) is a formula, A ⊆ M x is some set and b ∈ U y .Say that a formula ψ(x, z) over ∅ (with z a tuple of variables each of the same sort as x) is an honest definition of tp φ opp (b/A) if for every finite A 0 ⊆ A there is some c ∈ A z such In other words, for all a ∈ A, if ψ(a, c) holds then so does φ(a, b) and for all a ∈ A 0 the other direction holds: if φ(a, b) holds then ψ(a, c) holds.
It is proved in [Sim15a, Theorem 6.16], [CS15,Theorem 11] that if T is NIP then for every φ(x, y) there is a formula ψ(y, z) that serves as an honest definition for any type in S φ (A) provided that |A| ≥ 2 (by [CS15, Remark 16] only some NIP is required of φ and formulas expressing consistency of Boolean combinations of vc(φ) + 1 instances of φ).In this section we improve this by proving this result assuming only that φ is NIP.
Corollary 5.23.Let φ(x, y) be NIP.Then there exists ψ(x, z) such that if A ⊆ U x with |A| > 1 and b ∈ U y , then ψ(x, z) is an honest definition of tp φ opp (b/A). Namely, where n and k are as in Theorem 5.17 with d = vc(φ) and α = 1/2.
Proof.By Theorem 5.17, tp φ opp (b/A) is the rounded average of k-compressible types p 0 , . . ., p n−1 ∈ S φ opp (A).Now we proceed as in the proof of Lemma 3.20: if A 0 ⊆ fin A, there are 2 n, and for each such i < n, p i | Di ⊢ φ(a, y) (by choice of D i ), so ψ(a, d) holds.On the other hand, if ψ(a, d) holds, then clearly |{i < n | φ(a, y) ∈ p i }| > 1 2 n, hence φ(a, b) holds.
Remark 5.24.In fact, by a Löwenheim-Skolem argument, to prove Corollary 5.23 we require Theorem 5.17 only in the case that A is countable.The proof of this case of Theorem 5.17 is slightly simpler, in that we can use ω in place of P fin (κ) (using [Sim15b, Lemma 2.8] instead of [Sim15b, Lemma 2.9]), and the usual Ramsey theorem in place of Proposition 5.15.
Remark 5.25.If φ(x, y) is stable, one can use Theorem 5.17 similarly to get a new way to see definability of φ-types over arbitrary sets (since k-isolated types are definable). 5.5.Hypes.
Definition 5.26.Suppose φ(x, y) is a formula and A ⊆ U y .For k ∈ N, a k-hype4 in φ over A5 is a collection Γ of instances of φ and ¬φ over A such that: ( (2) For any a ∈ A, either φ(x, a) ∈ Γ or ¬φ(x, a) ∈ Γ, but not both.
Suppose π(x) is a (small) partial type.We say that Γ is k-consistent with π if in (1) we ask that S ∪ π is consistent.
Let S x φ,k (A) be the set of k-hypes in φ over A, and S π φ,k (A) the set of k-hypes which are k-consistent with π.
As with types, if Γ is a k-hype, we use the notation Γ| A ′ for the restriction of Γ to A ′ ⊆ A with the obvious meaning.
We deduce the existence of honest definitions for k-hypes.
Corollary 5.30.Let φ(x, y) be NIP and let k be as in Theorem 5.29 for d = vc(φ) and α = 1/2.Then there exists ψ(x, z) such that if A ⊆ U x with |A| > 1 and Γ ∈ S φ opp ,k (A) is a k-hype, then ψ(x, z) is an honest definition of Γ in the sense that if A 0 ⊆ fin A, then there is some d ∈ A z such that: (1) If a ∈ A 0 and φ(a, y) ∈ Γ then ψ(a, d) holds.
(2) For all a ∈ A, if ψ(a, d) holds, then φ(a, y) ∈ Γ. Namely, where n and k are as in Theorem 5.29.
Proof.The proof is the same as the one of Corollary 5.23, using Theorem 5.29.
We relate hypes to the Shelah expansion which we now recall.
Definition 5.31.For a structure M , the Shelah expansion M Sh of M is given by: for any formula φ(x, y) and any b ∈ U y , add a new relation Corollary 5.33.Suppose T is NIP, and let M T .For each formula φ(x, y), let k φ be as in Theorem 5.29 for d = vc(φ) and α = 1/2.Consider the expansion M hSh of M given by naming for each partitioned Lformula φ and each k φ -hype Γ ∈ S φ opp ,k φ (M x ) the set R Γ := {a ∈ M x | φ(a, y) ∈ Γ}.Then M hSh is interdefinable with M Sh and in particular is NIP.
Proof.Since every φ opp -type is in particular a k φ -hype, every R φ(x,b) (M ) is definable in M hSh .
Definition 5.34.Let L ′ = L ∪ {P, Q} where P, Q are predicates for subsets of U x .Suppose φ(x, y) is an L-formula, M T , D ⊆ M x , and p ∈ S φ opp (D).For ǫ < 2, let D ǫ = {a ∈ D | φ(a, y) ǫ ∈ p}.Then p is pseudofinite if for every L ′ -sentence ϕ, if (M, D 0 , D 1 ) ϕ then there is an L ′ -structure N such that N ϕ and P N , Q N are finite.
Remark 5.35.In the notation of Definition 5.34, a type p ∈ S φ opp (D) is pseudofinite iff there is a model (N, E 0 , E 1 ) ≡ (M, D 0 , D 1 ) which is an ultraproduct i∈I N i /U such that (E ǫ ) Ni is finite for each i ∈ I and ǫ < 2 (essentially the same proof as in [Vää03, Lemma 1] works).
Theorem 5.36.Suppose φ(x, y) is NIP.Then there is a formula ψ(x, z) such that whenever M T , D ⊆ M x is of size > 1 and p ∈ S φ opp (D) is pseudofinite, p is definable by an instance of ψ over D z .
Moreover, if T has Skolem functions, then we can choose ψ(x, z) to be NIP.
Proof.For the first part, let k and ψ(x, z) be as in Corollary 5.30.Suppose that p(y) is not definable by an instance of ψ over D z .Working in the expansion (M, D 0 , D 1 ) as in Definition 5.34, we get that in some L ′ -structure (N, E 0 , E 1 ), letting E = E 0 ∪ E 1 , the following hold: • E is finite of size > 1.
• The formula φ(x, y) is NIP in N and its VC-dimension equals vc(φ).
• The set of formulas Γ = {φ(a, y) not definable in N by any instance of ψ over E z .However, by the choice of ψ and as E is finite, there is some For the second part, assuming that T has Skolem functions, we let ) , where n is as in Theorem 5.29 and f is a ∅-definable function such that T thinks that if ∃y j<k (φ(z i,j , y) )) (whose existence we assumed).Note that ψ(x, z) is NIP, since φ(x, f (z)) is NIP; see also [EK20, Proof of Proposition 26].To see that it works, assume not.Then using the same argument as above, we get an L ′ -structure N with the same properties as above.Now, review the proof of Corollary 5.23.When the domain D of the k-compressible types p i (for i < n) is finite, then p i is in fact isolated by a conjunction of k instances of φ opp or its negation, thus, putting the isolating parameters for z i and coding the negations using z ′ i , z ′′ i and two elements from D N , we are done.Remark 5.37.Note that if p is realised in M , then Theorem 5.36 follows directly from UDTFS: in that case, in the proof one can replace the demand about Γ being a hype by it being a type.

Compressibility as an isolation notion
In this section we study properties of compressibility seen as an isolation notion (mostly) under NIP, and in particular as a way to construct models analogous to constructible models in totally transcendental theories.Towards that we prove a transitivity result for compressibility in Proposition 6.25, which uses the type decomposition theorem from [Sim20].
As an application, we will show that if T is unstable and M T is ω-saturated, then there are arbitrarily large elementary extensions N of M such that every generically stable type over M (see Definition 4.16) realised in N is realised in M (this is Corollary 6.39 and Remark 6.40).Proof.We start by showing that tp(c/Ab) is compressible.Suppose z is a tuple of variables such that b ∈ U z .Given a formula φ(x, y), let Φ be the set of all formulas of the form ψ(x, z, y) we get from substituting variables from y by variables from yz in φ.Fix some formula ψ(x, z, y) ∈ Φ.By assumption, there is some formula ζ ψ (xz, w) that compresses tp ψ (cb/A) (with the partition ψ(xz, y)).This means that for any A 0 ⊆ fin A there is some a ψ,A0 ∈ A w such that tp(cb/A) ⊢ ζ ψ (xz, a ψ,A0 ) ⊢ tp ψ (cb/A 0 ).
However, the converse to Lemma 6.1 holds for infinite tuples as well (see Remark 6.27 below).For finite tuples this can be seen by a direct argument of this kind, but for infinite tuples we will need stronger tools which we will develop in the next section under NIP.Definition 6.3.Suppose B, A are sets.We say B is compressible over A if tp( b/A) is compressible where b is some (any) tuple enumerating B.
Remark 6.4.The set B is compressible over B (even isolated).Remark 6.5.By Remark 3.17 B is compressible over A iff for every finite tuple b from B, tp(b/A) is compressible over A. Lemma 6.6.Given a set B and a tuple a, the set Ba is compressible over B if and only if tp(a/B) is compressible.
Proof.Left to right follows from Lemma 6.1, so suppose that tp(a/B) is compressible.Let b be a tuple enumerating B. We must show that tp(ab/B) is compressible.By Remark 3.17, it is enough to prove that tp(ab ′ /B) is compressible where b ′ is a finite sub-tuple of b.Let y be a tuple of variables in the sort of b ′ .Let φ(x, y, z) be a formula and let ψ(x, s) compress tp φ(x,yz) (a/B).Then ψ(x, s) ∧ y = t compresses tp φ(xy,z) (ab ′ /B).

Type decomposition and rescoping compressibility.
Here we use the results from [Sim20] to prove that compressibility can be rescoped to an arbitrary subset of the domain (see Propositions 6.18 and 6.23).
For the remainder of Section 6 we assume that T is NIP unless otherwise specified.
We first recall the definition of a generically stable partial type.As opposed to previous sections, here a partial type does not have to be small, i.e., it is over U. We call such partial types global partial types.As for global types, a global partial type π is A-invariant if it is invariant under automorphisms of U fixing A. Remark 6.7.Suppose π(x) is a global partial type.Then for any small set A, if a π| A , then π(x) ∪ tp(a/A) is consistent, and hence for any B there is some a ′ ≡ A a such that a ′ π| B .Definition 6.8.We say that a global partial type π is ind-definable over A if for every φ(x, y), the set {b ∈ U y | φ(x, b) ∈ π} is ind-definable over A, i.e., it is a union of A-definable sets.Remark 6.9.[Sim20, Discussion after Definition 2.1] Note that π(x) is ind-definable iff {φ(x, c) | π ⊢ φ(x, c)} is ind-definable.Fact 6.10.[Sim20, Lemma 2.2] Let π(x) be an A-invariant global partial type.Then π is ind-definable over A if and only if the set X = {(a, b) | b ∈ U ω , a π| A b} is type-definable over A. Definition 6.11.Let π(x) be a global partial type.We say that π is generically stable over A if π is ind-definable over A and the following holds: (GS) if (a k | k < ω) is such that a k π| Aa <k and π ⊢ φ(x, b), then for all but finitely many values of k we have U φ(a k , b).Remark 6.12.Note that a global type p(x) ∈ S x (U) is generically stable over A as in Definition 4.12 iff it is generically stable over A as a partial type (note that it is A-definable by Fact 4.13).
Remark 6.13.Much like in Remark 3.17, a global partial type π(x) is generically stable iff its restriction to any finite sub-tuple x ′ of x is generically stable.(Note that we do not assume that π is ind-definable.) Why? Clearly if the restrictions are all generically stable then π is, so we show the converse.Assume that π is generically stable and fix some finite x ′ ⊆ x.First note that π ↾ x ′ is ind-definable, so we show (GS).Assume that (a i and a i π| Aa<i for all i < n.Suppose we found such a sequence (a ) is a sequence of length n + 1 which is as required.By compactness and Fact 6.10, there is some sequence (a i | i < ω) such that a i ↾ x ′ = a ′ i and a i π| Aa<i for all i < ω.By (GS) for π, for all but finitely many values of k we have φ(a ′ k , b), as required.Definition 6.14.We say that a global partial type π(x) is finitely satisfiable in A ⊆ U if any formula implied by π has a realisation in A. We now state [Sim20, Theorem 4.1] in the form we will use it below.Our formulation follows from the proof (rather than the statement) of [Sim20, Theorem 4.1], in particular from [Sim20, Proposition 4.7].Fact 6.16.[Sim20, Proposition 4.7] Given a type tp(a/A) and q ∈ S A-fs (U), there exists a global partial type π(x) generically stable over A with a π| A such that if (X, <) is an infinite linear order, I q (X) | Aa , b q| AI and a π| AIb , then b q| Aa .Remark 6.17.(i) In [Sim20, Proposition 4.7] the generically stable type constructed depends on q.Call it π q .However, in the paragraph after [Sim20, Proposition 4.7], it is remarked that taking π to be the union of all the π q works for all q.(ii) It is not assumed in [Sim20] that the the tuple a above is finite.(iii) Throughout the proof of [Sim20, Theorem 4.1], the sequences are assumed to be densely ordered without endpoints.In particular, that is the case for I above.However, the result is true for any infinite I. Indeed, suppose I, a, A, b are as in Fact 6.16.By Ramsey and compactness (and as I is infinite) there is an Aab-indiscernible sequence I ′ = (a i ) i∈Q realising the EM-type of I over Aab.Since I is Aa-indiscernible, it follows that I ′ q (Q) | Aa , and since q is A-invariant, b q| AI ′ .Also, by Fact 6.10, a π| AI ′ b .So I ′ satisfies all the requirements of Fact 6.16 and is densely ordered with no endpoints, so b ⊢ q| Aa as required.(iv) In the context of Fact 6.16, it follows (by applying an automorphism; note that both q and π are A-invariant) that if a ′ π| AIb ∪ tp(a/AI), then b q| Aa ′ .Proposition 6.18.Let (X, <) be any (small) infinite linearly ordered set.A type tp(a/A) is compressible if and only if for every q ∈ S A-fs (U), if I q (X) | Aa , then Moreover, if X has no first element then q| AI ⊢ q| AIa .
Proof.That this condition implies compressibility follows from Fact 2.14(3⇒1), since I can be taken from A ′ (where (A ′ , a) is saturated enough).
For the converse, assume compressibility and let q, I be as in the proposition.Let d be given by Corollary 2.15 so that q| Ad ⊢ q| Aa and tp(d/Aa) is finitely satisfiable in A. By perhaps changing d (by applying an automorphism fixing Aa), we may assume that I q (X) | Aad .
Let We conclude the "moreover" part.Suppose that X has no first element and that I q (X) | Aa , b q| AI .By compactness we can find some I ′ of order type ω × (X + 1) (where X + 1 is adding one more element in the end of X and the product is ordered lexicographically), such that I ′ + I is indiscernible over Aa and over Ab.Thus, partitioning I ′ into (X + 1)-sequences, we have that I ′ (q (X+1) ) (ω) | Aa , and (I + b) q (X+1) | AI ′ .Applying the first part to q (X+1) , we have that (I + b) q (X+1) | Aa .It follows that b q| AIa as required.
The following corollary will not be used in this paper.Corollary 6.19.A type p = tp(a/A) ∈ S(A) is compressible iff for any q ∈ S A-fs (U) and any Morley sequence I := I 1 + (b) + I 2 of q over A where I 1 has no first element, if I 1 + I 2 is a Morley sequence of q over Aa then so is I.
Proof.Right to left is clear by Proposition 6.18, so suppose that p is compressible, and we are given I as above.Let (b i | 1 ≤ i < n) be some finite subsequence from I 2 and let b 0 = b.Then by applying the "moreover" part of Proposition 6.18 inductively, b i q| AI1b<ia .Since this is true for any n < ω, I is Aa-indiscernible.Indeed, to show the first statement, it is enough to see that p := tp(c/B) is compressible by Lemma 6.6.This is true since ζ(x, z) := (z < x) ∧ (x = 1) compresses it: for every finite B 0 ⊆ B, let i < ω be such that B 0 ≤ b i .Then p ⊢ (b i < x) ∧ (x = 1) ⊢ p| B0 by quantifier elimination.
Since every tuple from B is in the definable closure of A, B is compressible over A (every finite tuple is isolated).
Finally, C is not compressible over A, since compressibility is monotonic (Lemma 6.1) and c is not compressible over A. Why? Suppose ψ(x, z) compresses tp x≥y (c/A).Let A 0 = a <|z|+2 ⊆ A, and assume d ∈ A z is such that ψ(c, d) holds and ψ Note that this example shows that in T , weak compressibility is different from compressibility (see Question 3.23 for the definition).Namely, tp(c/A) is weakly compressible (as witnessed by ζ(x, z)) but not compressible.
Remark 6.27.The following rephrasing of Proposition 6.25 is worth mentioning explicitly: given (perhaps infinite) tuples c, b and a set A, if tp(c/Ab) and tp(b/A) are compressible, then tp(cb/A) is compressible.In this phrasing, this is a converse to Lemma 6.1 (see Remark 6.2).
This follows from Proposition 6.25 and Lemma 6.6.Note that Proposition 6.23 where B \ A is finite can be seen with a direct argument (not using NIP), so for finite tuples c, b, the above can be easily proven and does not require NIP.6.4.Compressible models and applications.In this section, T ′ is a countable NIP theory with monster model U in the language L ′ , F ⊆ U is some small subset, and T = Th(U F ) in the language L := L ′ (F ).In other words, T is a complete theory we get by naming constants in a countable NIP theory (whose monster model is still denoted by U, abusing notation).Definition 6.28.Say B is compressibly constructible over A if B can be enumerated as B = (b i ) i<α for some ordinal α, such that tp(b i /Ab <i ) is compressible for all i < α.
As with other isolation notions, the existence of compressibly constructible models follows straightforwardly from density.In fact this is an instance of the abstract result [She90, Theorem IV.3.1(5)], but we give the proof.Proposition 6.29.For any set A, there exists a model M ⊇ A which is compressibly constructible over A and of cardinality ≤ |A| + |T |.
Moreover, if B is compressibly constructible over A then there is some model M ⊇ B which is compressibly constructible over A and of cardinality ≤ |B| + |T |.
Proof.Since A is compressibly constructible over A, it is enough to prove the "moreover" part.
Let λ = |B|+|T |.Construct an increasing chain (B i ) i<ω as follows.Let B 0 = B, and given B i , let (θ i j ) j<λ enumerate all consistent formulas over B i .Construct a sequence (b i j ) j<λ inductively by letting b i j θ i j be such that tp(b i j /B i b i <j ) is compressible, using Corollary 6.24 (with B = F ). Let B i+1 = B i ∪ {b i j | j < λ}.Finally, M := i<ω B i is as required, by Tarski-Vaught.
Thanks to Proposition 6.25, we also have the following instance of [She90, Theorem IV.3.2(1)].Proposition 6.30.If B is compressibly constructible over A, then B is compressible over A.
Suppose c ∈ U. Then tp(c/M ) is generically stable (see Definition 4.16).Indeed, suppose c / ∈ M and let p be a global coheir extending tp(c/M ).Then any Morley sequence I := (a i ) i<ω of p over M must be such that a i , a j are incomparable for i = j.Thus I is an indiscernible set by quantifier elimination, and hence p is generically stable by NIP and the "moreover" part of Fact 4.13.
Thus, if tp(c/M ) is compressible then it is l-isolated by Proposition 4.14 and hence realised by Remark 4.7.Hence M is the only model compressible over M and the conclusion of Corollary 6.37 does not hold.Corollary 6.39.Let M T , and suppose that < is an M -definable preorder on an M -definable set D and suppose that there is an infinite <-chain in D(M ).Then: (i) There exists N ≻ M with D(N ) D(M ) such that if a ∈ N and tp(a/M ) is generically stable, then a ∈ M .(See Definition 4.16.)(ii) For any cardinal λ there is N ≻ M with |D(N )| ≥ λ such that if a ∈ N and tp(a/M ) is generically stable, then a ∈ M .In particular, there are arbitrarily large elementary extensions N of M such that if S is M -definable and stable (see Definition 4.4), then S(M ) = S(N ).
Proof.(ii) follows from (i) by taking the union of a suitably long elementary chain.We prove (i).
By Remark 4.9 there is some c ∈ D(U) such that tp(c/M ) is compressible but not l-isolated.In particular c / ∈ M .Apply (the "moreover" part of) Proposition 6.29 to get some compressibly constructible model N ≻ M over M containing M c.By Proposition 4.14, if a ∈ N and tp(a/M ) is generically stable, then it is l-isolated and hence by Remark 4.7, it is realised.
The "in particular" part follows since for any a ∈ S(U), tp(a/M ) is stable and hence generically stable by Fact 4.18.Remark 6.40.Note that the condition in Corollary 6.39 holds whenever T is unstable and M is ω-saturated, since in this case T has the SOP ([Sim15a, Theorem 2.67]) and hence any ω-saturated model contains an infinite chain in some definable preorder.

Compressible types in ACVF.
Example 6.41.If U ACVF and A ⊆ U eq , then any model M containing A whose residue field is algebraic over A is compressible over A; i.e., if M ≺ U, A ⊆ M eq , and k(M ) = acl eq (A) ∩ k, where k = k(U) denotes the residue field, then M is compressible over A.
To see this, consider first a 1-type tp(a/B) over an acl eq -closed set B = acl eq (B) ⊆ U eq , where a ∈ U. We show that tp(a/B) is compressible iff dcl eq (Ba) ∩ k = B ∩ k.By unique Swiss cheese decompositions [Hol95, Theorem 3.26], tp(a/B) is determined by knowing which of the balls defined over B contain a, i.e., tp(a/B) is implied by tp ∈ (a/B) where x ∈ y is the element relation between the valued field and the sort of balls (closed and open).One sees directly that this ∈-type is compressible unless tp(a/B) is the generic type of a closed ball α over B which contains infinitely many open balls over B of radius γ := radius(α).In that case, let β 1 = β 2 be distinct open subballs of α over B of radius γ (the existence of these two balls is our only use of our assumption that there are infinitely many such), and consider the map β → res( β−β1 β2−β1 ) for β an open subball of α of radius γ.This is a well-defined injective map to k defined over B, so genericity of tp(a/B) (and the fact that B = acl eq (B)) implies that the image under this map of the open ball around a of radius γ is not in B ∩ k, so we have dcl eq (Ba) ∩ k B ∩ k.Conversely, if tp(a/B) is compressible, then since the residue field is a pure stably embedded algebraically closed field, we must have dcl eq (Ba) ∩ k = B ∩ k.
The compressibility of M over A if k(M ) = acl eq (A) ∩ k now follows by first taking a compressible construction sequence which alternates taking acl eq -closure and adding a single new element of M , and then applying Proposition 6.30.
If A ∩ k is infinite then conversely k(M ) = acl eq (A) ∩ k is necessary for compressibility of M over A, but this fails in general; for example, if A is finite, then any model containing A is of course compressible over A.
Remark 6.42.The argument in Example 6.41 is based on ideas from [BM21], and combining it with [BM21, Remark 3.12] actually yields an alternative proof of [BM21, Theorem 5.6].Indeed, suppose K is a valued subfield of U with res(K) finite.Taking an elementary extension we may assume that (U, K) is sufficiently saturated.Let A ⊆ K. Then k(K alg ) = acl eq (A) ∩ k is the algebraic closure of the prime field, so by Example 6.41, K alg , and in particular K, is compressible over A. By [BM21, Remark 3.12], it follows that K is distal in U in the sense defined in that paper.The same argument applies to K r defined in [BM21, Theorem 5.6].However, this proof does not yield bounds on the exponents of the resulting distal cell decompositions, whereas such bounds are obtained in [BM21, Remark 6.2] as a consequence of the more elementary methods of that paper.

Proof.
Let s > B vc (n, k), and let A 0 ⊆ U x with |A 0 | = s.For each i, by Sauer-Shelah [Sim15a, Lemma 6.4], at most s ≤k subsets of A 0 are defined by instances of φ i ; hence at most s ≤k n are defined by instances of θ.It follows from the definition of B vc (n, k) that θ does not shatter A 0 .2.4.Compressible types.Here we will review the basic properties of compressible types.

Theorem 3. 4 .
For any d < ω, let k comp (d) := 2 d+1 (d − 2) + d + 4. For any set A, if C ⊆ 2 A is closed, non-empty, and has VC-dimension ≤ d, then there exists c ∈ C which is k comp (d)-compressible in C. Proof.The proof is by induction on d.If vc(C) = 0, then C is a singleton {c}, and c is clearly 0-compressible in C. Suppose that vc(C) = d + 1 > 0. Claim 3.5.Let k 0 := 2 d+1 d + 1.There is D ⊆ A and c ′ ∈ C| D which is k 0compressible in C| D such that vc(C c ′ ) ≤ d.
In fact we may take D 1 with |D 1 | = k 0 , since |D| ≥ k 0 by maximality and the assumption that |A| ≥ k 0 .Since vc(C) ≤ d+1 = |B| and 2 B ⊆ C c ′ , for each a ∈ D 1 there is e a ∈ 2 B such that e a ⊢ C c ′ | {a} .By the choice of k 0 = 2 d+1 d + 1 and the pigeonhole principle, there exist e ∈ 2 B and E ⊆ D 1 such that |E| = d + 1 and e 3 there is a cofinal monochromatic subset, yielding e ∈ 2 B which is as required.Now by the induction hypothesis there is c ∈ C c ′ which is k comp (d)-compressible in C c ′ .We conclude by showing that c is (k comp (d) + k 0 )-compressible in C; this gives the stated bound, since (2 d+1 (d − 2) + d + 4) + (2 d+1 d + 1) = 2 (d+1)+1 ((d + 1) − 2) + (d + 1) + 4. So suppose A 0 ⊆ fin A and let A 1 ⊆ A be such that |A 1 | ≤ k comp (d) and c| A1 ⊢ C c ′ c| A0 .By compactness of C it follows that there is a finite subset D 0 ⊆ D such that c| A1 ⊢ C c ′ | D 0 c| A0 .Let D 1 be such that |D 1 | ≤ k 0 and c ′ | D1 ⊢ C|D c ′ | D0 (which exists as c ′ ∈ S).Then c| A1∪D1 ⊢ C c| A0 , as required.
Lemma 4.3.Let φ(x, y) be NIP.Then the following are equivalent:(i) φ is stable.(ii)For any B ⊆ U y and p ∈ S φ (B) and k ∈ N, if p is k-compressible then p is isolated (and hence k-isolated by Remark 4.2).(iii) For all k < ω and ǫ ∈ {0, 1} k , θ ǫ (x, y) := j<k φ(x, y j ) ǫj does not have the strict order property.
Fact 4.5.[EK21, Remark 2.6] The following are equivalent for a partial type π(x): and b <i are already defined, let b i be such that p ∋ ζ(x, b i ) ⊢ (p ↾ φ)| a<i (x), and let a i be such that ζ(x, b j ) ⊢ (p ↾ φ)| ai (x) for all j ≤ i, which exists since j≤i ζ(x, b j ) ⊢ (p ↾ φ)(x).(i.b)For ψ(x) ∈ p, we show that θ(x, z) := ζ(x, z) ∧ ψ(x) has the order property by recursively constructing (a i , b i , c i ) i<ω such that θ(a i , b j ) iff i ≥ j and j<i θ(x, b j ) ∈ p and φ(a i , c i ) ⇔ φ(x, c i ) / ∈ p.This is enough by Fact 4.5.Suppose we found (a j , b j , c j ) j<i .Let b i be such that p ∋ ζ(x, b i ) ⊢ (p ↾ φ)| c<i (x).Since

Fact 4. 13 .
Suppose p is generically stable over A. Then: (1) [PT11, Proposition 2.1] p is A-definable and finitely satisfiable in every model containing A. (2) [PT11, Proposition 2.1] If I is a Morley sequence of p over A then I is an indiscernible set (i.e., totally indiscernible).
Definition 4.16.Suppose M T .A type p ∈ S(M ) is generically stable if it has a global M -invariant extension which is generically stable over M .Remark 4.17.If p ∈ S(M ) is generically stable then it has a unique M -invariant extension by [PT11, Proposition 2.1(iii)].Note that if p ∈ S(M ) then it has a global M -invariant extension (e.g., a coheir).Thus, together with [CG20, Proposition 3.4], we get the following fact.
Fact 4.18.If M is a model and p ∈ S(M ) is stable then p is generically stable.It follows that when the base is a model, Lemma 4.8(i.b) is implied by Proposition 4.14.

Fact 6. 15 .
Let π(x) be a global partial type generically stable over A. Then: (FS) π is finitely satisfiable in every model containing A. (NF) Let φ(x, b) be such that π ⊢ φ(x, b) and take a π| A such that ¬φ(a, b).Then both tp(b/Aa) and tp(a/Ab) fork over A.
Remark 6.20.One might call the condition in Corollary 6.19 generic co-distality: it is co-distality in a generic sense.For a definition of distal and co-distal types and a short discussion, see [EK21, Definition 4.21 and Remark 4.22].Definition 6.21.Suppose B, A ⊆ U are (small) sets.Say that a type p ∈ S(A) is compressible up to B if p is compressible in U B (in the language L(B)): in Definition 3.14, all the formulas are over B. Remark 6.22.Suppose that A ⊆ B, p ∈ S(A) and p is compressible up to B. Then p is compressible up to C := B \ A: given a formula φ(x, y) over C, there is an L-formula ψ(x, z, w, t) and a ∈ A w , c ∈ C t such that ψ(x, z, ac) compresses p ↾ φ.But then ψ(x, z, w, c) compresses p ↾ φ.Proposition 6.23.If a type tp(a/B) is compressible and A ⊆ B, then tp(a/A) is compressible up to B.
b q| AI .We want to show that b q| Aa .Applying Fact 6.16 (and Remark 6.17(ii),(iv)) to tp(d/A), we obtain a generically stable over A global partial type π with d π| A such that b q| Ad ′ for any d ′ π| AIb ∪ tp(d/AI).Now tp(d/Aa) is finitely satisfiable in, and hence does not fork over, A. Similarly, tp(I/Aad) is finitely satisfiable in A and so does not fork over Aa.By applying (NF) twice, it follows that d π| AIa .Hence π ′ := π ∪ tp(d/AIa) is consistent by Remark 6.7.Let κ := |L(AIb)| + , and let (d i ) i<κ be a Morley sequence in π ′ over AIa, i.e., d i π ′ | AIad<i .By (GS) and the choice of κ, for some i < κ we have d i π| AIb .Then d i π| AIb ∪ tp(d/AI), so b q| Adi .But d i ≡ Aa d, so q| Adi ⊢ q| Aa .So b q| Aa , as required.