Characterizations of monadic NIP

We give several characterizations of when a complete first-order theory $T$ is monadically NIP, i.e. when expansions of $T$ by arbitrary unary predicates do not have the independence property. The central characterization is a condition on finite satisfiability of types. Other characterizations include decompositions of models, the behavior of indiscernibles, and a forbidden configuration. As an application, we prove non-structure results for hereditary classes of finite substructures of non-monadically NIP models that eliminate quantifiers.


Introduction
It is well known that many first-order theories whose models are tame can become unwieldy after naming a unary predicate. Arguably the best known example of this is the field (C, +, ·) of complex numbers. Its theory is uncountably categorical, but after naming a predicate for the real numbers, the expansion becomes unstable. A more extreme example is the theory T of infinite dimensional vector spaces over a finite field, in a relational language. The theory T is totally categorical, but if, in some model V , one names a basis B, then by choosing specified sum sets of basis elements, one can code arbitrary bipartite graphs in expansions of V by unary predicates.
As part of a larger project in [2], Baldwin and Shelah undertook a study of this phenomenon. They found that a primary dividing line is whether T admits coding i.e., there are three subsets A, B, C of a model of T and a formula φ(x, y, z) that defines a pairing function A × B → C. If one can find such a configuration in a model M of T , some monadic expansions of M are wild. The primary focus in [2] was monadically stable theories, i.e. theories that remain stable after arbitrary expansions by unary predicates. Clearly, the two theories described above are stable, but not monadically stable. They offered a characterization of monadically stable theories within the stable theories via a condition on the behavior of nonforking. This allowed them to prove that monadic stability yields a dividing line within stable theories: models of monadically stable theories are well-structured and admit a nice decomposition into trees of submodels, while if a theory is stable but not monadically stable then it encodes arbitrary bipartite graphs in a unary expansion, and so is not even monadically NIP.
A theory T is NIP if it does not have the independence property, and is monadically NIP if every expansion of a model of T by unary predicates is also NIP. The behavior of NIP theories has been extensively studied, see e.g., [13]. Soon after [2], Shelah further studied monadically NIP theories in [11], where he showed they satisfy a condition on the behavior of finite satisfiable types paralleling the condition on the behavior of non-forking in monadically stable theories. He was then able to use this to produce a linear decomposition of models of monadically NIP theories, akin to a single step of the tree decomposition in monadically stable theories.
We dub Shelah's condition on the behavior of finite satisfiability the f.s. dichotomy, and we consider it to be the fundamental property expressible in the original language L describing the dichotomous behavior outlined above. We show the f.s. dichotomy characterizes monadically NIP theories and provide several other characterizations, including admitting a linear decomposition in the style of Shelah, a forbidden configuration, and conditions on the behavior of indiscernible sequences after adding parameters. Definitions for the following theorem may be found in Definitions 3.9, 3.1, 3.4, and 3.8. Of note is that all but the first two conditions refer to the theory T itself, rather than unary expansions. Theorem 1.1. The following are equivalent for a complete theory T with an infinite model.
(3) T does not admit coding on tuples. (4) T has the f.s. dichotomy. We believe that monadic NIP (or perhaps a quantifier-free version) is an important dividing line in the combinatorics of hereditary classes, and provides a general setting for the sort of decomposition arguments common in structural graph theory. For example, see the recent work on bounded twin-width in the ordered binary case, where it coincides with monadic NIP [4,15]. Here, we mention the following conjecture, adding monadic NIP to a question of Macpherson [9,Question 2.2.7]. Conjecture 1. The following are equivalent for a countable homogeneous ω-categorical relational structure M .
(2) The (unlabeled) growth rate of Age(M ) is at most exponential.
From Theorem 1.1, we see that if T is not monadically NIP then it admits coding on tuples. This allows us to prove the following non-structure theorem in Section 5 (with Definition 5.1 defining the relevant terms), in particular confirming (2) ⇒ (1) and a weak form of (3) ⇒ (1) from the conjecture, although without any assumption of ω-categoricity.

Theorem 1.3. Suppose T is a complete theory with quantifier elimination in a relational language with finitely many constants. Then Age(T ) is NIP if and only if T is monadically NIP, and Age(T ) is stable if and only if T is monadically stable.
In Section 2, we review basic facts about finite satisfiability, and introduce M -f.s. sequences, which are closely related to, but more general than Morley sequences. The results of this section apply to an arbitrary theory, and so may well be of interest beyond monadic NIP. Section 3 introduces the f.s. dichotomy and proves the equivalence of (3)-(6) from Theorem 1.1. Much of these two sections is an elaboration on the terse presentation of [11], although there are new definitions and results, particularly in Subsection 3.2, which deals with the behavior of indiscernibles in monadically NIP theories. In Section 4 we finish proving the main theorem by giving a type-counting argument that the f.s. dichotomy implies monadic NIP, and by showing that if T admits coding on tuples then it admits coding in a unary expansion. In Section 5, we prove Theorems 1.2 and 1.3.
We are grateful to Pierre Simon, with whom we have had numerous insightful discussions about this material. In particular, the relationship between monadic NIP and indiscernible-triviality was suggested to us by him.
1.1. Notation. Throughout this paper, we work in C, a large, sufficiently saturated, sufficiently homogeneous model of a complete theory T . We routinely consider tp(A/B) when A is an infinite set. To make this notion precise, we (silently) fix an enumerationā of A (of ordinal order type) and an enumerationx with lg(x) = lg(ā). Then tp(A/B) = {θ(x ,b) : C |= θ(ā ,c) for all subsequencesx ⊆x andā is the corresponding subsequence ofā}.

M -f.s. sequences
Forking independence and Morley sequences are fundamental tools in the analysis of monadically stable theories in [2]. These are less well-behaved outside the stable setting, but in any theory we may view 'tp(A/M B) is finitely satisfiable in M ' as a statement that A is (asymmetrically) independent from B over M . Following [11], we will use finite satisfiability in place of non-forking, and indiscernible M -.f.s. sequences in place of Morley sequences. Throughout Section 2, we make no assumptions about the complexity of T h(C).

Preliminary facts about
One way of producing finitely satisfiable types in M comes from average types.
It is easily checked that Av(U, B) is a complete type over B that is finitely satisfied in M . We record a few basic facts about types that are finitely satisfied in M . Proofs can be found in either Section VII.4 of [12] or in [13].
Note that for any C ⊇ M , A i : i ∈ I is an M -f.s. sequence over C if and only if the concatenation C A i : i ∈ I is an M -f.s. sequence. We note two useful operations on M -f.s. sequences over C, 'Shrinking' and 'Condensation'. (1) 'Shrinking:' For every J ⊆ I, for all A j ⊆ A j , and for all C with C ⊇ C ⊇ M , we say A j : j ∈ J as a sequence over C is obtained by shrinking from A i : i ∈ I as a sequence over C. (2) 'Condensation:' Suppose π : I → J is a condensation, i.e., a surjective map with each π −1 (j) a convex subset of I. For each j ∈ J, let A * j := { A i : i ∈ π −1 (j) }. We say A * j : j ∈ J as a sequence over C is obtained by condensation from A i : i ∈ I as a sequence over C.
In particular, removing a set of A i 's from the sequence is an instance of Shrinking.
Proof. As notation, choose disjoint sets {x i : i ∈ I } of variables, with lg(x i ) = lg(A i ) for each i ∈ I. For each i ∈ I, choose an ultrafilter For a finite, non-empty t = { i 1 < i 2 < · · · < i n } ⊆ I, letx t =x i 1 . . .x i n . We will recursively define complete types w t (x t ) ∈ Sx t (D) as follows: That is,ā t realizes w t if and only ifā s realizes w s and, for every θ( It is easily checked that each w t (x t ) is a complete type over D and, arguing by induction on |t|, whenever t ⊆ t, w t is the restriction of w t tox t . Thus, by compactness, w * := { w t (x t ) : t ⊆ I non-empty, finite } is consistent, and in fact, is a complete type over D. Choose any realization A i : i ∈ I of w * . Then, for each i ∈ I, tp(A i /DA <i ) = Av(U i , DA <i ). Since D ⊇ C and tp(A i /CA <i ) = Av(U i , CA <i ), it follows that tp( A i : i ∈ I /C) = tp( A i : i ∈ I /C). Thus, it suffices to choose any D satisfying A i : i ∈ I D ≡ C A i : i ∈ I D .
2.2. C ⊇ M full for non-splitting. Definition 2.11. We call C ⊇ M full (for non-splitting over M ) if, for every n, every p ∈ S n (M ) is realized in C.
The relevance of fullness is that, whenever C ⊇ M is full, every complete type q ∈ S(C) has a unique extension to any set D ⊇ C that does not split over M . Keeping in mind finite satisfiability as an analogue of non-forking, the next lemma says that 'types over C that are finitely satisfied in M are stationary.'  Proof. Left to right is obvious. For the converse, we need to show that tp(B 2 /B 1 AC) is finitely satisfied in M . To begin, by Proposition 2.10, choose As tp(B 2 /B 1 AC) finitely satisfied in M , so is tp(B 2 /B 1 AC). Lemma 2.14. Suppose C ⊇ M is full and A, B /C is an M -f.s. sequence over C. Choose anyā 1 ,ā 2 from A andb 1 ,b 2 from B with tp(ā 1 /C) = tp(ā 2 /C) and tp(b 1 /C) = tp(b 2 /C). Then tp(ā 1b1 /C) = tp(ā 2b2 /C).
We glean two results from Lemma 2.14. The first bounds the number of types realized in an M -f.s. sequence, independent of either |I| or |A i |. The second is a refinement of the type structure of an M -f.s. sequence over a full C ⊇ M .

M -f.s. sequences and indiscernibles.
In this subsection, we explore the relation between M -f.s. sequences and indiscernibles. An M -f.s. sequence need not be indiscernible (for example, the tuples can realize different types), but when it is, it gives a special case of a Morley sequence in the sense of [13].
We first show indiscernible sequences can always be viewed as M -f.s. sequences over some model M .

Lemma 2.18 (extending [11, Part I Lemma 4.1]). Suppose (I, ≤) is infinite and
Proof. Right to left is clear, so assume I is both indiscernible over M and an M -f.s. sequence. As (I, ≤) is infinite, it contains either an ascending or descending ω-chain. For definiteness, choose J ⊆ I of order type ω. To ease notation, we writeā k in place ofā j k . For each k ∈ ω and each formula φ( Finally, as J ⊆ I and I is indiscernible over M , an easy induction on lg(b) gives the result.
Using Lemma 2.19, we obtain a strengthening of Lemma 2.18. The lemma below can be proved by modifying the proof of Lemma 2.10, but the argument here is fundamental enough to bear repeating. Proof. For the first sentence, given I, M and C, choose an ultrafilter U as in Lemma 2.19. A routine compactness argument shows that we can find a sequence By contrast, if C ⊇ M and A i : i ∈ I /C is an M -f.s. sequence over C, then (2) is satisfied, but (1) may fail. In the case where C ⊇ M is full, (1) reduces to a question about types over C.
Proof. Left to right is clear. For the converse, fix i < j. By Lemma 2.21, it suffices to show A j realizes p i . But this is clear, as both tp In terms of existence of such sequences, we have the following.
Proof. By compactness it suffices to prove this for (I, ≤) = (ω, ≤). By Fact 2.3(1), choose an ultrafilter U on M lg(x) and recursively letā i be a realization of Av(U, Cā <i ). It is easily checked that ā i : i ∈ ω /C is an M -f.s. sequence over C with tp(ā i /C) = p for each i. As C ⊇ M is full, it is also indiscernible over C by Lemma 2.22.
3. The f.s. dichotomy We begin this section with the central dividing line of this paper. Although unnamed, the concept appears in Lemma II 2.3 of [11]. It would be equivalent to replaceā,b by sets A, B ⊂ C in the definition above, and this form will often be used. Much of the utility of the f.s. dichotomy is via the following extension lemma. ≤) is a well-ordering with a maximum element, we may take J = I.

Lemma 3.2 ([11, Part I Claim 2.4]). Suppose T has the f.s. dichotomy and
Proof. Fix any M -f.s. sequence A i : i ∈ I and choose any singleton c ∈ C. Let I 0 ⊆ I be the maximal initial segment of I such that tp(c/A I 0 M ) is finitely satisfied in M . Note that I 0 could be empty or all of I. If the minimum element of (I \ I 0 ) exists, name it i * and take J = I; otherwise, let J = I ∪ {i * }, where i * is a new element realizing the cut (I 0 , I\I 0 ) and put A i * = ∅. Let We first show that tp To finish, we show that for every j > i * , tp(A j /cA <j M ) is finitely satisfied in M . Again, if this were not the case, the f.s. dichotomy would imply tp(cA j /A <j M ) is finitely satisfied in M . But then, by Shrinking, we would have tp(c/A <j M ) finitely satisfied in M , contradicting our choice above.
For the 'Moreover' sentence, the only concern is if tp(c/A I M ) is finitely satisfied in M . But in this case we may take i * to be the maximal element of I, rather than a new element in J\I.
In [2,Theorem 4.2.6], the f.s. dichotomy appears as a statement about the behavior of forking rather than non-forking. Namely, forking dependence is totally trivial and transitive on singletons. We may derive similar consequences for dependence from the f.s. dichotomy. This is stated in [3,Corollary 5.22], although missing the necessary condition of full C. (2) By induction on lg(ā). Left to right is immediate, so assume tp(a/Mb) is finitely satisfied in M for every a ∈ā. Writeā =ā a * . By induction we may assume tp(ā /Mb) is finitely satisfied in M . By the f.s. dichotomy, either tp(ā a * /Mb) is finitely satisfied in M and we are done immediately, or else tp(ā /Mba * ) is finitely satisfied in M , and we finish using transitivity from Fact 2.3.
(3) Left to right is immediate by Shrinking, so assume tp(a/Cb) is finitely satisfied in M for every a ∈ā and b ∈b. It follows from (2) that tp(ā/Cb) is finitely satisfied in M for every b ∈b. To conclude that tp(ā/Cb) is finitely satisfied in M , we argue by induction on lg(b). Letb =b b * , and by induction assume the statement is true forb .
By the f.s. dichotomy, either tp(ā/Mb b * ) or tp(ā b * /Mb ) is finitely satisfiable in M . In the first case we are finished immediately, and in the second we finish by invoking Lemma 2.13.
In the stable case, forking dependence is symmetric as well and so yields an equivalence relation on singletons, which is used in [2] to decompose models into trees of submodels. In general, the f.s. dichotomy shows finite satisfiability yields a quasi-order on singletons when working over a full C ⊇ M . Taking the classes of this quasi-order in order naturally gives an irreducible decomposition of C over M in the sense of the next subsection, but we sometimes wish to avoid having to work over a full C ⊇ M .
3.1. Decompositions of models. In this subsection, we characterize the f.s. dichotomy in terms of extending partial decompositions to full decompositions of models.
By iterating Lemma 3.2 for every c ∈ X for a given set X we obtain: Remark 3.7. We could do the above proof over some full C ⊇ M to obtain an irreducible M -f.s. decomposition that is also an order-congruence.
We note that at the end of [2], Baldwin and Shelah conjecture that models of monadically NIP theories should admit tree decompositions like those they describe for monadically stable theories, but with order-congruences in place of full congruences.

Preserving indiscernibility.
We begin with some definitions. The definition of dp-minimality given here may be non-standard, but it is proven equivalent to the usual definition with Fact 2.10 of [6]. Definition 3.8 (Indiscernible-triviality and dp-minimality). The first definition is meant to recall trivial forking.
• T has indiscernible-triviality if for every infinite indiscernible sequence I and every set B of parameters, if I is indiscernible over each b ∈ B then I is indiscernible over B. • T is dp-minimal if, for all indiscernible sequences I = ā i : i ∈ I over any set C, every b ∈ C induces a finite partition of the index set into convex pieces I = I 1 I 2 · · · I n , with at most two I j infinite and every I j = ā i : i ∈ I j is indiscernible over Cb.
As mentioned in the introduction, the notion of a theory admitting coding was the central dividing line of [2]. We weaken the definition here to allow the sequences to consist of tuples. Note that even the theory of equality would admit the further weakening of also allowing C to consist of tuples.
T admits coding if we may take I and J to be sequences of singletons.
A convenient variant for this subsection is a joined tuple-coding configuration, which consists of a formula (with parameters) φ(x,ȳ, z), a sequence ā i : i ∈ I indiscernible over the parameters of φ, indexed by an infinite linear order (I, ≤), and a set { c i,j | i < j ∈ I } such that for i < j, C |= φ(ā i ,ā j , c k,l ) ⇐⇒ (i, j) = (k, l). Given a joined tuple-coding configuration, indexed by a countable, dense (I, ≤), we may construct a tuple-coding configuration by keeping φ(x,ȳ, z) fixed, choosing open intervals I , J ⊆ I with I J , and letting I = ā i : i ∈ I and J = ā j : j ∈ J . Conversely, given a tuple-coding configuration with I = J, we may construct a joined tuple-coding configuration by considering the indiscernible sequence whose elements are ā ibi : i ∈ I , restricting C to elements c i,j with i < j, and replacing φ by φ * (xx ,ȳȳ , z) := φ(x,ȳ , z).
The following configuration appears in [11, Part II Lemma 2.2], and will appear as an intermediate between a failure of the f.s. dichotomy and a tuple-coding configuration.

Definition 3.10.
A pre-coding configuration consists of a φ(x,ȳ, z) with parameters and a sequence I = d i : i ∈ Q , indiscernible over the parameters of φ, such that for some (equivalently, for every) We show the equivalence of the existence of these notions with the proposition below. The proof of (4) ⇒ (1) in the following is essentially from [11, Part II Lemma 2.3], while (3) ⇒ (4) is based on [15,Lemma C.1]. The idea of (4) ⇒ (1) is that when working over a full D ⊇ M , types have a unique "generic" extension by Lemma 2.12. In a failure of the f.s. dichotomy, the extension of tp(c/D) to tp(c/Dab) is non-generic, and so c can in some sense pick out a and b from a suitable sequence.
Proposition 3.11. The following are equivalent for any theory T .
(2) T is dp-minimal and has indiscernible-triviality. Using this, we argue that I is indiscernible over MB in two steps. First, we argue that I is indiscernible over MB 0 . To see this, fix i < j from Q and let p i = tp(ā i /A <i MB 0 ). From the previous paragraph, I is an M -f.s. sequence over MB 0 . So p i does not split over M , and so by Lemma 2.21 it suffices to prove that a j realizes p i . Choose any φ(x,b,m) ∈ p i withm from M andb from B 0 . To see that C |= φ(ā j ,b,m), choose an automorphism σ ∈ Aut(C) fixing M and an initial segment ā i : i ∈ I 0 pointwise that induces an order-preserving permutation of I with σ(ā i ) =ā j . Clearly, C |= φ(ā j , σ(b),m). It is easily seen that for every singleton b ∈ σ(b), I is indiscernible over Mb and, as σ fixes A I 0 pointwise, b is also low. Thus, any simple extension to ( I)MB 0 σ(b) will condense to B 0 σ(b) ā i : i ∈ Q B 1 . In particular tp(ā j /Mbσ(b)) is finitely satisfied in M . Thus, if C |= ¬φ(ā j ,b,m), by finite satisfiability there would ben from M such that C |= ¬φ(n,b,m) ∧ φ(n, σ(b),m), which is impossible since σ fixes M pointwise. Thus, I is indiscernible over MB 0 .
Finally, to see that I is indiscernible over MB 0 B 1 , choose any i 1 < . . . i k , j 1 < · · · < j k from Q,b from MB 0 , andc from B 1 and assume by way of contradiction that C |= ψ (ā i 1 , . . . ,ā i k ,b,c) ∧ ¬ψ(ā j 1 , . . . ,ā j k ,b,c). Recall B 0 , I, B 1 is an M -f.s. sequence, so tp(c/M ( I)B 0 ) is finitely satisfied in M , and so the same formula is true with somem from M replacingc. But this contradicts that I is indiscernible over MB 0 . Thus, I is indiscernible over MB.
Ifā i ,ā j are in the same convex piece, then taking i < k < j we get φ(ā k ,ā j , c i,j ), contradicting our configuration. So supposeā i andā j are in different pieces. Then one of the pieces must be infinite, so by symmetry suppose the piece I containinḡ a i is. By indiscernible-triviality (ā i : i ∈ I ) is indiscernible overā j c i,j . But then picking some k ∈ I \ { i } again gives φ(ā k ,ā j , c i,j ).
Remark 3.12. The following observation will be useful in Section 5. A tidy pre- But if e ∈d k for k ≤ i then the former type is finitely satisfiable in M , and if e ∈d k for k > i, then the latter type is.
The tidiness property extends to the joined tuple-coding configuration constructed in (3) ⇒ (4) and so ultimately to the tuple-coding configuration as well. That is, from a failure of the f.s. dichotomy, we construct a tuple-coding configuration I, J , C, φ with C |= ¬φ(ā i ,b j , e) for everyā i ∈ I,b j ∈ J , and e ∈ I ∪ J .

The main theorem
We recall the main theorem from the introduction. Note that whereas Clauses (1) and (2)   The equivalences of (3)-(6) are by Proposition 3.6 and Proposition 3.11. We note that (1) ⇒ (2) is easy: Choose a monadic expansion C * that admits coding, say via an L-formula φ(x, y, z) defining a bijection from the countable sets A × B → C. By adding a new unary predicate for a suitable C 0 ⊆ C, the formula ψ(x, y) := ∃z ∈ C 0 φ(x, y, z) can define the edge relation of an arbitrary bipartite graph on A × B, and in particular of the generic bipartite graph. Thus, T is not monadically NIP.

4.1.
If T has the f.s. dichotomy, then T is monadically NIP. The typecounting argument in this section is somewhat similar to that in [3], showing that monadic NIP corresponds to the dichotomy of unbounded partition width versus partition width at most 2 (ℵ 0 ). Both arguments use the tools from Sections 2 and 3 to decompose the model and count the types realized in a part of the decomposition over its complement. However, while Blumensath decomposes the model into a large binary tree, our decomposition takes a single step.  We will be primarily interested in the case where A is very large, and rtp(N, A) is significantly smaller than |A|. The following lemma is similar to Lemma 2.15, removing the requirement that the partition is convex but adding a finiteness condition.
For the rest of this section, recall the notation A J = j∈J A j from the first part of Definition 2.4.  On the other hand, if a theory T has the independence property, then no uniform bound can exist. Proof. In the monster model, choose an order-indiscernible I = (a i : i ≤ λ) that is shattered, i.e., there is a set Y = {b s : s ∈ P(λ) } such that φ(b s , a i ) holds if and only if i ∈ s. Note that for distinctb,b ∈ Y , there is some a i ∈ I such that tp φ (b/a i ) = tp φ (b /a i ). Let N be any model containing I ∪ Y and let A i : i ≤ λ be any I-partition of N . As |I| = λ, while |Y | = 2 λ , by applying the pigeon-hole principle n times (one for each coordinate ofb) one obtains Y ⊆ Y , also of size 2 λ , and a finite I 0 ⊆ I such thatb ∈ (A I 0 ) n for eachb ∈ Y . As λ > 2 |T | and there are at most 2 |T | types over I 0 , we can find Y * ⊆ Y of size 2 λ such that tp(b/I 0 ) is constant amongb ∈ Y * . It follows that tp(b/( To show that the behaviors of Lemma 4.4 and Lemma 4.5 cannot co-exist, we get an upper bound on the number of types realized in a finite monadic expansion. Such a bound is easy for quantifier-free types, and the next lemma inductively steps it up to a bound on all types. The following two lemmas make no assumptions about T . For each k ∈ ω, define an equivalence relation ∼ k on (N \ A) <ω by:ā ∼ kb if and only if lg(ā) = lg(b) and tp φ (ā/A) = tp φ (b/A) for every formula φ(z) of quantifier depth at most k. Clearly, tp(ā/A) = tp(b) if and only ifā ∼ kb for every k. To get an upper bound on rtp (N, A), for each k ∈ ω, let r k (N, N, A)).
Proof. The second sentence follows from the first as tp(ā/A) = tp(b/A) if and only ifā ∼ kb for every k. For the first sentence, we give an alternate formulation of ∼ k to make counting easier. For each k ∈ ω, let E k be the equivalence relation on (N \ A) <ω given by: •  N, A) and by the definition of E k+1 we have c(k + 1) ≤ 2 c(k) for each k, so the lemma follows from the fact that E k (ā,b) if and only ifā ∼ kb , whose verification amounts to proving the following claim.

Lemma 4.7. Let N ⊇ A be any model and let
Proof. For each n, expanding by k unary predicates can increase the number of quantifier-free n-types by at most a finite factor, i.e. 2 k , so r 0 (N + , A) = r 0 (N, A) ≤ rtp (N, A). The result now follows from Lemma 4.6.
Finally, we combine the lemmas above to obtain the goal of this subsection.

Proposition 4.8. If
T has the f.s. dichotomy, then T is monadically NIP.
Proof. By way of contradiction assume that T is not monadically NIP, but has the f.s. dichotomy. Let T + be an expansion by finitely many unary predicates that has IP. Choose a cardinal λ > ω+1 (|T |). Let N + |= T + with N + ⊇ I = (a i : i ≤ λ) as in Lemma 4.5, so for any I-partition of N + there is Let N be the L-reduct of N + . As I remains L-order-indiscernible, and T has the f.s. dichotomy, choose an I-partition A i : I ≤ λ of N as in Lemma 4.4, so rtp(N, A J ) ≤ 2 (|T |) for every J ⊆ (λ + 1). Since N + is a unary expansion of N , rtp(N + , A J ) ≤ ω+1 (|T |) for every J ⊆ (λ + 1), by Lemma 4.7. This contradicts our ability to find an I 0 ⊆ (λ + 1) from the previous paragraph for the chosen I-partition of N + . Lemma 2.15 and the arguments in this subsection seem to indicate that, for a generalization of the structural graph-theoretic notion of neighborhood-width [7] similar to Blumensath's generalization of clique-width [3], monadic NIP should correspond to a dichotomy between bounded and unbounded neighborhood-width.

From coding on tuples to coding on singletons.
This subsection provides the final step, (2) ⇒ (3), in proving Theorem 4.1 by showing that if T admits coding on tuples, then some monadic expansion admits coding (i.e., on singletons). For the result of this subsection, since T admitting coding on tuples immediately implies T is not monadically NIP, we could finish by [2,Theorem 8.1.8], which states that if T has IP then this is witnessed on singletons in a unary expansion. But the number of unary predicates used would depend on the length of the tuples in the tuple-coding configuration, which would weaken the results of Section 5.
Deriving non-structure results in a universal theory from the existence of a bad configuration is made much more involved if the configuration can occur on tuples. If one is willing to add unary predicates, arguments such as that from [2] mentioned above will often bring the configuration down to singletons. A general result in this case is [3,Theorem 4.6] that (under mild assumptions) there is a formula defining the tuples of an indiscernible sequence in the expansion adding a unary predicate for each "coordinate strip" of the sequence. The results of [14] indicate the configuration can often be brought down to singletons just by adding parameters, instead of unary predicates, but these arguments seem difficult to adapt to tuplecoding configurations. Another approach, which we use here, is to take an instance of the configuration where the tuples have minimal length, and argue that the tuples then in many ways behave like singletons. A tuple-coding configuration as above is regular if wheneverd ⊆ A,ē ⊆ B (including cases withd ∈ I,ē ∈ J ), σ is a standard permutation of A corresponding to an element of Aut(I, ≤) fixing i, and τ is a standard permutation of B corresponding to an element of Aut(J, ≤) fixing j.
By Ramsey and compactness, if T admits coding on tuples via the formula φ(x,ȳ, z), then it admits a regular tuple-coding configuration via the same formula φ(x,ȳ, z). A tuple-coding configuration has unique witnesses up to permutation if for every c i,j ∈ C, the only witnesses for c i,j are of the form (σ(ā i ), τ (b j )) for some σ a permutation ofā i and some τ a permutation ofb j ȳ, z) be a regular tuple-coding configuration for T , with |x| + |ȳ| minimal. Then this configuration has unique witnesses up to permutation.
Let I * ⊆ I be an open interval such thatd ∩ (ā i : i ∈ I * ) =d * . Let φ * (x * ,ȳ, z) be the formula obtained by starting with φ(d,ȳ, z), and then replacing the subtuplē d * with the variablesx * ; so we have plugged the elements ofd\d * as parameters into φ. For each k ∈ I * , letā * k be the restriction ofā k to the coordinates corresponding tod * . Then ā * is also a regular tuple-coding configuration, contradicting the minimality of |x| + |ȳ|.
The following Lemma completes the proof of Theorem 4.1.

Lemma 4.12. Suppose T admits coding on tuples. Then T admits coding in an expansion by three unary predicates.
Proof. Choose a tuple-coding configuration with |x| + |ȳ| as small as possible. By the remarks following Definition 4.9. we may assume this configuration is regular, so by Lemma 4.11, it has unique witnesses up to permutation. Let L * = L ∪ { A, B, C } and let C * be the expansion of C interpreting A as Ā , B as B , and C as itself. Let → ¬φ(xx , yȳ , z ))).
Let a i be the first coordinate ofā i , and b j the first coordinate ofb j . Then A 1 = { a i : i ∈ I } , B 1 = { b j : j ∈ J }, and C witness coding in T * = T h(C * ) via the L * -formula φ * (x, y, z).

Finite structures
In this section, we restrict the language L of the theories we consider to be relational (i.e., no function symbols) with only finitely many constant symbols. The growth rate of Age(T ) (sometimes called the profile or (unlabeled) speed) is the function ϕ T (n) counting the number of isomorphism types with n elements in Age(T ).
We also investigate Age(T ) under the quasi-order of embeddability. We say Age(T ) is well-quasi-ordered (wqo) if this class does not contain an infinite antichain, and we say Age(T ) is n-wqo if Age(M * ) is wqo for every expansion M * of any model M of T by n unary predicates that partition the universe.
The definition of n-wqo is sometimes given for an arbitrary hereditary class C rather than an age, with C n-wqo if the class C * containing every partition of every structure of C by at most n unary predicates remains wqo. Our definition is possibly weaker, but then its failure is stronger.

Example 1. Let T = T h(Z, succ). Then Age(T ) is wqo, but not 2-wqo, since
Age(T ) contains arbitrarily long finite paths, and marking the endpoints of these paths with a unary predicate gives an infinite antichain.
By contrast, if T = T h(Z, ≤), then Age(T ) can be shown to be n-wqo for all n.
The following lemma shows that when considering n-wqo, adding finitely many parameters is no worse than adding another unary predicate. Proof. Suppose an expansion by k constants is not n-wqo, as witnessed by an infinite antichain { M + i } i∈ω in a language L + expanding the initial language by the k constants and by n unary predicates. Let M * i be the structure obtained from M + i by forgetting the k constants, but naming their interpretations by a single new unary predicate. As Age(M ) is (n + 1)-wqo, { M * i } i∈ω contains an infinite chain As there are only finitely many permutations of the constants, some embedding in the chain must preserve them, contradicting that { M + i } i∈ω is an antichain. In both Theorems 5.3 and 5.6, the assumption that T has quantifier elimination is only used to get that the formula witnessing that T admits coding on tuples is quantifier-free, and the formula witnessing the order property in the stability part of Theorem 5.6, so the hypotheses of the theorems can be weakened to only these specific formulas being quantifier-free. This weakened assumption is used in [15]. From the proof of Proposition 3.11, if the failure of the f.s. dichotomy is witnessed by quantifier-free formulas, then the formula witnessing coding on tuples will be quantifier-free as well. Theorem 5.3. If a complete theory T has quantifier elimination in a relational language with finitely many constants is not monadically NIP, then Age(T ) has growth rate asymptotically greater than (n/k)! for some k ∈ ω and is not 4-wqo.
Proof. Since T is not monadically NIP, let φ(x,ȳ, z) be a regular tuple-coding configuration with unique witnesses up to permutation. The only place we use T has QE is to choose φ quantifier-free. Let L * expand by unary predicates for A, B, and C as well as constants for the parameters of φ, and let φ * be as in the proof of 4.12. Let A ⊆ Age(T * ) be the set of finite substructures that can be constructed as follows.
( Proof of Claim. Since φ is quantifier-free, it remains to check that if the existential quantifiers in φ * are witnessed in C and C |= φ * (a, b, c) then they are witnessed in M , and if the universal fails in C then it fails in M . From the unary predicates at the beginning of φ * , we may let a ∈ā i , b ∈b j , and c = c k, . If C |= φ(a, b, c), the only tuple in C that can witnessx is the rest of the tupleā i , which will be in M because it only contains full tuples, and similarly for witnessingȳ . Since our configuration has unique witnesses up to permutation, if the universal quantifier fails in C, this is witnessed by an element c k , with i − ≤ k ≤ i + and j − ≤ ≤ j + . By regularity, this failure is also witnessed by some element in { c i± ,j , c i,j± }. ♦ Given a bipartite graph G with n edges and no isolated vertices, we may encode it as a structure M G ∈ A by starting with tuplesā i for each point in one part and tuplesb j for each point in the other part, and including c k, whenever we want to encode an edge betweenā k andb . Note that |M G | = O(n), and this encoding preserves isomorphism in both directions. In the proof of [8, Theorem 1.5], the asymptotic growth rate of such graphs is shown to be at least (n/5)!, which gives the desired growth rate for Age(T * ) with the constant k depending on the length of the tuples in the tuple-coding configuration. Since expanding by finitely many unary predicates and constants increases the growth rate by at most an exponential factor, we also get the desired growth rate for Age(T ). Furthermore, if M H embeds into M G , then H must be a (possibly non-induced) subgraph of G. So we get that Age(T * ) is not wqo by encoding even cycles. We expanded by three unary predicates, and by Lemma 5.2 the parameters may be replaced by another unary predicate while still preserving the failure of wqo, so we get that Age(T ) is not 4-wqo.
Remark 5.4. There is a homogeneous structure, with automorphism group S ∞ Wr S 2 in its product action, that is not monadically NIP and whose growth rate is the number of bipartite graphs with a prescribed bipartition, n edges, and no isolated vertices. So the lower bound in this theorem cannot be raised above the growth rate of such graphs. Precise asymptotics for this growth rate are not known, although it is slower than n! and [5, Theorem 7.1] improves Macpherson's lower bound to ( n log n 2+ ) n for every > 0. If Conjecture 1 from the Introduction (in particular (1) ⇒ (2)) is confirmed, then the lower bound on the growth rate in Theorem 5.3 would also confirm [8, Conjecture 3.5] that for homogeneous structures there is a gap from exponential growth rate to growth rate at least (n/k)! for some k ∈ ω.
Theorem 5.3 is somewhat surprising. Since passing to substructures can be simulated by adding unary predicates, it is clear that if T is monadically tame, then Age(T ) should be tame. However, unary predicates can do more, so it seems plausible that Age(T ) could be tame even though T is not monadically tame. Our next theorem gives some explanation for why this does not occur, at least when assuming quantifier elimination.
First we need to define stability and NIP for hereditary classes. The following definition is standard and appears, for example, in A class of structures C has IP if there is some formula φ(x,ȳ) such that for every finite, bipartite graph G = (I, J, E), there is some M G ∈ C encoding G via φ. Otherwise, C is NIP.
A class of structures C is unstable if there is some formula φ(x,ȳ) such that for every finite half-graph G, there is some M G ∈ C encoding G via φ. Otherwise, C is stable.
Equivalently, by compactness arguments, C is NIP (resp. stable) if and only if every completion of T h(C), the common theory of structures in C, is. Note that it suffices to witness that C has IP or is unstable using a formula with parameters, since we can remove them by appending the parameters to eachā i .
The sort of collapse between monadic NIP and NIP in hereditary classes observed in Theorem 5.6 occurs for binary ordered structures [15], since there the formula giving coding on tuples is quantifier-free. It also occurs for monotone graph classes (i.e. specified by forbidding non-induced subgraphs), where NIP actually collapses to monadic stability, and agrees with nowhere-denseness [1].

Theorem 5.6. Suppose that a complete theory T in a relational language with finitely many constants has quantifier elimination. Then Age(T ) is NIP if and only if T is monadically NIP, and Age(T ) is stable if and only if T is monadically stable.
Proof. We first consider the NIP case.
(⇐) Suppose Age(T ) has IP, as witnessed by the formula φ(x,ȳ). By compactness, there is a model N of the universal theory of T in which φ encodes the generic bipartite graph. But then N is a substructure of some M |= T , and naming a copy of N in M by a unary predicate U and relativizing φ to U gives a unary expansion of M with IP.
(⇒) Suppose T is not monadically NIP, witnessed by a tuple-coding configuration I = (ā i : i ∈ I), J = (b j : j ∈ J), C = { c i,j | i ∈ I, j ∈ J } , φ(x,ȳ, z), with φ quantifier-free and containing parametersm. By Remark 3.12, we may also assume the configuration is tidy. For any bipartite graph G, let M G ∈ Age(T ) contain m, tuples from I and J corresponding to the two parts of G, and an element of c i,j for each edge of G so that R * (x,ȳ;m) := ∃z ∈ C(φ(x,ȳ, z;m)) encodes G on I(M G ) × J (M G ). But by tidiness, R(x,ȳ;m) := ∃z(φ(x,ȳ, z;m) ∧ z ∈m) encodes G on I(M G ) × J (M G ) as well.
For the stable case, the backwards direction is the same except using the infinite half-graph in place of the generic bipartite graph. For the forwards direction, if T is unstable then by quantifier-elimination Age(T ) is also unstable. If T is stable but not monadically stable, then by [2, Lemma 4.2.6] T is not monadically NIP, so we are finished by the NIP case.