On uniform definability of types over finite sets for NIP formulas

TL;DR

This paper proves that a formula is NIP if and only if it satisfies uniform definability of types over finite sets, settling a conjecture and linking model theory with machine learning concepts.

Contribution

It establishes the equivalence between NIP and UDTFS for formulas, confirming a conjecture and advancing understanding of model-theoretic properties.

Findings

Proves NIP iff UDTFS for formulas

Settles Laskowski's conjecture

Links model theory with machine learning theory

Abstract

Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets (UDTFS). This settles a conjecture of Laskowski.

Equations

$$\exists d_0,...,d_{m-1}\underset{t<m}{\bigwedge }\underset{i\in s_t}{\bigwedge }\phi (d_t,z_i{)}^{{\epsilon }_{i,t}}\wedge \underset{t\in s}{\bigwedge }\phi (d_t,y)$$

$$\forall d_0,...,d_{m-1}\underset{t<m}{\bigwedge }\underset{i\in s_t}{\bigwedge }\phi (d_t,z_i{)}^{{\epsilon }_{i,t}}\to \underset{t\in s}{\bigwedge }\phi (d_t,y)$$

$$a_i\in \phi (b,A)\iff M\models \phi (b,a_i)\iff$$

$$M\models \phi (f(\overline{a},{\overline{\epsilon }}^{(\overline{a},b)}),a_i)\iff a_i\in \phi (f(\overline{a},{\overline{\epsilon }}^{(\overline{a},b)}),A),$$

$$\left|\left\{t<m|M\models (\phi (c,a)\leftrightarrow \phi (f_{|\overline{a_t}|}(\overline{a_t},{\overline{\epsilon }}^{(\overline{a_t},c)}),a))\right\}\right|>\frac{m}{2}.$$

$$B_{i,j}=\{\begin{array}{ll}1& M\models (\phi (c,a_i)\leftrightarrow \phi (h_j,a_i))\\ 0& \text{else}\end{array}.$$

$${\mu }^{t}B{v}^j={\mu }^t\stackrel{\text{the }j\text{’th- column}}{\left(\begin{array}{c}|\\ B_j\\ |\end{array}\right)}=\sum _{i<|A|,B_{i,j}=1}\mu (a_i)=\sum _{a\in {\tilde{A}}_j}\mu (a)=$$

$$\mu (\left\{a\in A|M\models (\phi (c,a)\leftrightarrow \phi (h_j,a))\right\})\ge \frac{2}{3}.$$

$$\underset{q\in {△}^n}{\min }\underset{p\in {△}^m}{\max }{q}^{t}Mp=\underset{p\in {△}^m}{\max }\underset{q\in {△}^n}{\min }{q}^{t}Mp.$$

$${\left(u^i\right)}^{t}B\nu \ge \frac{2}{3}\Rightarrow \underset{\text{the i’th row}}{\left(\begin{array}{ccc}-& B_i& -\end{array}\right)}\nu \ge \frac{2}{3}\Rightarrow \sum _{h_{\overline{a}}\in \widetilde{{\mathcal{H}}_i}}\nu (h_{\overline{a}})\ge \frac{2}{3}$$

$$\nu (\left\{h_{\overline{a}}\in \mathcal{H}|M\models (\phi (c,a)\leftrightarrow \phi (h_{\overline{a}},a))\right\})\ge \frac{2}{3}.$$

$$\left|p(X\backslash Y)-\frac{|F\cap (X\backslash Y)|}{|F|}\right|\le ϵ.$$

$$\left|\nu (\neg \phi (\mathcal{H},a))-\frac{|F\cap \neg \phi (\mathcal{H},a)|}{|F|}\right|\le \frac{1}{8}.$$

$$\left|\left\{t<m|\phi (x,a)\in p\iff M\models \phi (f_{|\overline{a_t}|}(\overline{a_t},{\overline{\epsilon }}^{(\overline{a_t},c)}),a)\right\}\right|>\frac{m}{2}.$$

$$\left|\left\{t<m|\phi (x,a)\in p\iff M\models \phi (h_t,a)\right\}\right|>\frac{m}{2}.$$

$$\psi (y)=\exists d_0,...,d_{m-1}\underset{t<m}{\bigwedge }\underset{i<k}{\bigwedge }\phi (d_t,a_i^t{)}^{{\epsilon }^{(a_i^t,h_t)}}\wedge \left(\left|\left\{t<m|\phi (d_t,y)\right\}\right|>\frac{m}{2}\right)$$

$$M\models \underset{t<m}{\bigwedge }\underset{i<k}{\bigwedge }\phi (h_t,a_i^t{)}^{{\epsilon }^{(a_i^t,h_t)}}\wedge \left(\left|\left\{t<m|\phi (h_t,a)\right\}\right|>\frac{m}{2}\right).$$

$$\psi (y)=\exists d_0,...,d_{m-1}\underset{t<m}{\bigwedge }\underset{i<k}{\bigwedge }\phi (d_t,a_i^t{)}^{{\epsilon }^{(a_i^t,h_t)}}\wedge \left(\left|\left\{t<m|\phi (d_t,y)\right\}\right|>\frac{m}{2}\right)$$

$$\underset{s\subseteq m,|s|>m/2}{\bigvee }\exists d_0,...,d_{m-1}\underset{t<m}{\bigwedge }\underset{i<k}{\bigwedge }\phi (d_t,a_i^t{)}^{{\epsilon }^{(a_i^t,h_t)}}\wedge \underset{t\in s}{\bigwedge }\phi (d_t,y).$$

$$M\models \exists d_0,...,d_{m-1}\underset{t<m}{\bigwedge }\underset{i<k}{\bigwedge }\phi (d_t,a_i^t{)}^{{\epsilon }^{(a_i^t,h_t)}}\wedge \underset{t\in s}{\bigwedge }\phi (d_t,a)$$

$$M\models \forall d_0,...,d_{m-1}\underset{t<m}{\bigwedge }\underset{i<k}{\bigwedge }\phi (d_t,a_i^t{)}^{{\epsilon }^{(a_i^t,h_t)}}\to \underset{t\in s}{\bigwedge }\phi (d_t,a).$$

$${\psi }^{\prime }(y)=\underset{s\subseteq m,|s|>m/2}{\bigvee }\forall d_0,...,d_{m-1}\underset{t<m}{\bigwedge }\underset{i<k}{\bigwedge }\phi (d_t,a_i^t{)}^{{\epsilon }^{(a_i^t,h_t)}}\to \underset{t\in s}{\bigwedge }\phi (d_t,y)$$

$${\theta }_{\psi }(x)=\neg \exists z\in P(\forall y\in P(\phi (x,y)\leftrightarrow \psi (y,z)))$$

$$\phi (x,a)\in p\iff \left|\left\{t<m|\phi (f_{|\overline{a_t}|,{\overline{\epsilon }}^{(\overline{a_t},c)}}(\overline{a_t}),a)\right\}\right|>\frac{m}{2},$$

Full text

The second author would like to thank the Israel Science Foundation for partial support of this research (grants no. 1533/14 and 1254/18).

1. Introduction

Let $L$ be any language and let $T$ be any $L$-theory. An $L$-formula $\varphi(x,y)$ has uniform definability of types over finite sets (UDTFS) in $T$ iff there is a formula $\psi(y,z)$ which uniformly (in any model of $T$) defines $\varphi$-types over finite sets of size $\geq 2$ (see Definition 4). If $\varphi$ has UDTFS, then for any finite $A\subseteq M^{y}\vDash T$, the number of $\varphi$-types over $A$ is bounded by $\left|A\right|^{\left|z\right|}$, which immediately implies that $\varphi$ has finite VC-dimension in $T$, i.e., $\varphi$ is NIP (if $\varphi$ shatters a finite set $A$, then the number of $\varphi$-types over $A$ is exponential in $\left|A\right|$). This raises the question, asked by Laskowski, of whether these two notions (UDTFS and NIP) are equivalent. Note that in that case, this also implies the Sauer-Shelah lemma in the sense of counting types (see [She90, Chapter II, Theorem 4.10(4)]). See also the discussion in [LS13].

This question was first addressed in [JL10] where it was proved assuming that $T$ is weakly o-minimal. Later, [Gui12] extended this result to dp-minimal theories. Finally, in [CS15, Theorem 15] it was proved in the level of the theory $T$: a (complete) theory is NIP iff every formula has UDTFS. They actually proved something stronger: in NIP theories, every formula has uniform honest definitions. See Section 3.2 below for the definition.

The main theorem in this paper solves Laskowski’s question (and thus answers all the questions in the final paragraph of [Gui12]).

Main Theorem 1**.**

The following are equivalent for an $L$-theory $T$ and an $L$-formula $\varphi(x,y)$.

1. (1)

$\varphi$* is NIP in $T$ (i.e., NIP in any completion of $T$).* 2. (2)

$\varphi$* has UDTFS in $T$.*

(see Theorem 14 below.)

The proof has two ingredients, both from machine learning theory.

The first is [MY16] which proves the existence of* sample compression schemes *for concept classes of finite VC-dimension $d$ whose sizes are bounded in terms of $d$ (answering a question of Littlestone and Warmuth). Roughly speaking, this result says that there is some number $k$ depending only on $d$ such that for any finite set of labeled examples (concepts), it is possible to recover our knowledge on any concept by considering a specific subset of size $k$ and some bounded extra information. We do not use the result but rather its proof, and most importantly the proof of Claim 3.1 from there, which we translate to our language.

The second ingredient is [CCT16] where an upper bound for the* recursive teaching dimension* (RTD) is given for concept classes of finite VC-dimension $d$ (the bound in [CCT16] is exponential in $d$ and was later improved to a quadratic bound in [HWLW17]). Roughly speaking this means that there is some number $t$ (depending only on $d$) such that every concept can be identified by at most $t$ samples according to the recursive teaching model. See Fact 11 for a precise statement which follows by reading the definitions. This results translates in our language to the existence of $\varphi$-types which are isolated by their restriction to a set of bounded size (see Corollary 13). This result (or rather, its proof) will be used in a forthcoming work with Martin Bays and Pierre Simon which deals with “compressible” types in NIP theories.

Despite the fact that our proof is based on these two results, we do not need to define any of the machine learning notions mentioned above so that the proof can be read by anyone with a basic understanding of model theory.

Remark 2*.*

The first ingredient was known by experts for quite some time now (it was brought to our attention by Pierre Simon in 2015). It was known that it alone implies UDTFS assuming that the theory has definable Skolem functions (see Section 3.1.2).

We became aware of the second ingredient during the aforementioned work with Martin Bays and Pierre Simon, thanks to Nati Linial who answered our question about it.

The paper is organized as follows: Section 2 contains all the preliminaries and the proof of the main theorem. Section 3 contains some open questions.

Acknowledgement*.*

We thank the anonymous referee for their remarks, and specifically for suggesting to add Remark 12 which certainly helps the presentation.

2. Proof of the main theorem

Throughout fix a language $L$; all formulas, theories and structures will be in $L$. For notation, we use $\varphi(x,y)$ to denote a formula $\varphi$ with a partition of (perhaps a superset of) the free variables of $\varphi$. When $x$ is a tuple of variables and $A$ is a set, we write $A^{x}$ to denote the tuples of length $|x|$ from $A$; alternatively, one may think of $A^{x}$ as the set of functions from the tuple $x$ to $A$.

Definition 3**.**

Let $M$ be a $L$-structure and let $\varphi(x,y)$ be a formula. Suppose that $p(x)$ is a $\varphi$-type over a set $A\subseteq M^{y}$. We say that a formula $\psi(y)$ (not necessarily over $\emptyset$) defines $p$ if for all $b\in A$ we have $M\vDash\psi(b)\Leftrightarrow\varphi(x,b)\in p(x)$.

Definition 4**.**

(UDTFS) Let $T$ be an $L$-theory. We say that $\varphi(x,y)$ has *uniform definability of types over finite sets in $T$ *(UDTFS) if there exists a formula $\psi(y,z)$ such that for every $M\vDash T$ and all finite sets $A\subseteq M^{y}$ with $\left|A\right|\geq 2$ the following holds: for every $p(x)\in S_{\varphi}(A)$ there exist $c\in A^{z}$ such that $\psi(y,c)$ defines $p(x)$.

Definition 5**.**

(VC-dimension) Let $X$ be a set and $\mathcal{F}\subseteq\SS(X)$. Given $A\subseteq X$, we say that it is shattered by $\mathcal{F}$ if for every $S\subseteq A$ there is $F\in\mathcal{F}$ such that $F\cap A=S$. A family $\mathcal{F}$ is said be a VC-class if there is some $n<\omega$ such that no subset of $X$ of size $n$ is shattered by $\mathcal{F}$. In this case the VC-dimension of $\mathcal{F}$, denoted by $VC(\mathcal{F})$, is the smallest integer $n$ such that no subset of $X$ of size $n+1$ is shattered by $\mathcal{F}$.

Definition 6**.**

($\varphi$ is NIP in $T$) Suppose that $T$ is an $L$-theory and that $\varphi\left(x,y\right)$ is a formula. Say that *$\varphi(x,y)$ is NIP in $T$ *if for every $M\vDash T$, the family $\left\{\varphi(M,a)\,\middle|\,a\in M^{y}\right\}$ is a VC-class. This is equivalent to saying that $\varphi$ is NIP in any completion of $T$.

Remark 7*.*

Note that $\varphi$ is NIP in $T$ iff there is a bound $n<\omega$ such that for every $M\models T$, the VC-dimension of the family $\left\{\varphi(M,a)\,\middle|\,a\in M^{y}\right\}$ is bounded by $n$ (and this is first-order expressible). This follows by compactness. Denote the minimal such $n$ by $VC_{T}(\varphi)$ or just $VC(\varphi)$ if $T$ is clear from the context. From now on we will just write NIP for NIP in $T$ when $T$ is clear from the context.

Fact 8**.**

[Adl08*, Remark 9]*Suppose that $T$ is an $L$-theory. The family of formulas $\varphi\left(x,y\right)$ which are NIP is closed under finite Boolean combinations. Moreover, (it follows from the proof that) if $\varphi_{0},\varphi_{1}$ are such and $\psi$ is a Boolean combination of $\varphi_{0},\varphi_{1}$, then $VC_{T}(\psi)$ is bounded in terms of $VC_{T}(\varphi_{0}),VC_{T}(\varphi_{1})$.

Fact 9**.**

[Adl08*, Proposition 2]*If $\varphi(x,y)$ is NIP then so is $\varphi^{opp}(y,x)$ where $\varphi^{opp}$ is the formula $\varphi$ but with the partition of variables switched. As above, $VC_{T}(\varphi^{opp})$ is bounded in terms of $VC_{T}(\varphi)$, in fact $VC_{T}(\varphi^{opp})<2^{VC_{T}(\varphi)+1}$.

Fact 10**.**

[Sim15, Corollary 6.9]**(VC-theorem; the existence of $\epsilon$-approximations) For any $d<\omega$ and $0<\epsilon$ there is some $N=N(d,\epsilon)<\omega$ such that for any finite set $X$, for any $C\subseteq\SS(X)$ of VC-dimension $\leq d$ and every probability measure $\mu$ on $X$ there exists a multiset $Y\subseteq X$ of size $\left|Y\right|\leq N$ such that for all $s\in C$, $\left|\mu(s)-\frac{\left|Y\cap s\right|}{\left|Y\right|}\right|\leq\epsilon$.

The next fact is the “second ingredient” mentioned in the introduction.

Fact 11**.**

[CCT16, Theorem 3]**[HWLW17, Theorem 6]** For all $n<\omega$ there is some $t=t\left(n\right)$ such that if $X$ is a finite set and $\mathcal{F}\subseteq\SS(X)$ is a family of VC-dimension $\leq n$, then there is some $F\in\mathcal{F}$ and $X_{0}\subseteq X$ of size $|X_{0}|\leq t$ such that for all $F^{\prime}\in\mathcal{F}$, $F=F^{\prime}$ iff $X_{0}\cap F^{\prime}=X_{0}\cap F$.

Remark 12*.*

Even though we do not need the proof of Fact 11, we will provide a sketch, based on the proof of [CCT16, Lemma 4]. The proof is by induction on $n$. For $n=0$ this is trivial since then $\left|\mathcal{F}\right|=1$ (we can choose $t=0$). Assume the statement is true for $n-1$. Let $k=2^{n}\left(n-1\right)+1$ and let $t\left(n\right)=t\left(n-1\right)+k$. We may assume that $\left|X\right|\geq k$. Find $b,c$ such that $c\subseteq b\subseteq X$, $\left|b\right|=k$ and $\mathcal{F}_{b,c}=\left\{F\in\mathcal{F}\,\middle|\,F\cap b=c\right\}$ is nonempty of minimal size. We claim that $VC\left(\mathcal{F}_{b,c}\right)<n$ (considering $\mathcal{F}_{b,c}$ as a subset of $\SS\left(X\right)$). If not, then there is some $e\subseteq X$ such that $\left|e\right|=n$ which is shattered by $\mathcal{F}_{b,c}$. Note that $e\cap b=\emptyset$. For every $x\in b$ there is some subset $e_{x}\subseteq e$ such that if $F_{1},F_{2}\in\mathcal{F}$ and $F_{1}\cap e=e_{x}=F_{2}\cap e$ then $x\in F_{1}$ iff $x\in F_{2}$: indeed, otherwise for some $x\in b$, $\left\{x\right\}\cup e$ is shattered by $\mathcal{F}$, contradicting $VC\left(\mathcal{F}\right)\leq n$. By pigeonhole (and by the choice of $k$), we can find some $e^{\prime}\subseteq e$ and a subset $b^{\prime}\subseteq b$ such that $\left|b^{\prime}\right|=n$ and if $F_{1},F_{2}\in\mathcal{F}$ are such that $F_{1}\cap e=e^{\prime}=F_{2}\cap e$ then $F_{1}\cap b^{\prime}=F_{2}\cap b^{\prime}$ (which must be $c\cap b^{\prime}$). Let $b^{\prime\prime}=\left(b\cup e\right)\backslash b^{\prime}$ and $c^{\prime\prime}=\left(c\cup e^{\prime}\right)\backslash b^{\prime}$. Then $\emptyset\neq\mathcal{F}_{b^{\prime\prime},c^{\prime\prime}}\subsetneq\mathcal{F}_{b,c}$, a contradiction. Thus $t\left(n\right)=t\left(n-1\right)+k$ works: given $F,X_{0}$ as in the fact for $\mathcal{F}_{b,c}$, the pair $F,X_{0}\cup b$ will work.

Corollary 13**.**

Let $m,n<\omega$. Then there is some $k=k(n,m)$ such that if $T$ is an $L$-theory, $\varphi\left(x,y\right)$ is NIP and $VC(\varphi)\leq n$ then for every $M\vDash T$ and finite $A\subseteq M^{y}$ the following holds:

If $\chi(x)=\bigwedge_{i<m}\varphi(x,a_{i})^{\varepsilon_{i}}$ where for every $i<m$, $a_{i}\in A$ and $\varepsilon_{i}<2$ (in general, $\varphi^{0}=\varphi$ and $\varphi^{1}=\neg\varphi$) is consistent, then there is a type $p_{0}\in S_{\varphi}(A)$ and $A_{0}\subseteq A$ of size $|A_{0}|\leq k$ such that $p_{0}|_{A_{0}}\vdash p_{0}\vdash\chi(x)$.

Proof.

Let $k=k(n,m)=t(2^{n+1}-1)+m$ where $t(n)$ is from Fact 11. Let $A^{\prime}=A\backslash\left\{a_{i}\,\middle|\,i<m\right\}$, and consider the family $\mathcal{F}=\left\{\varphi(b,A^{\prime})\,\middle|\,b\vDash\chi\right\}$. Then the VC-dimension of $\mathcal{F}$ is bounded by the dual VC-dimension of $\varphi$ which is $\leq 2^{n+1}-1$. By Fact 11 there is some $b\in M^{x}$ and some $X_{0}\subseteq A$ of size $\leq t(2^{n+1}-1)$ such that $\chi(b)$ holds and for all $b^{\prime}\vDash\chi$, $\varphi(b,A^{\prime})=\varphi(b^{\prime},A^{\prime})$ iff $\varphi(b,X_{0})=\varphi(b^{\prime},X_{0})$. Let $p_{0}=\operatorname{tp}_{\varphi}\left(b/A\right)$ and let $A_{0}=X_{0}\cup\left\{a_{i}\,\middle|\,i<m\right\}$. Obviously $p_{0}\vdash\chi(x)$ and if $b^{\prime}\vDash p_{0}|_{A_{0}}$ then $b^{\prime}\vDash\chi$ and since $\varphi(b,X_{0})=\varphi(b^{\prime},X_{0})$, it follows that $\varphi(b,A^{\prime})=\varphi(b^{\prime},A^{\prime})$, so that $\varphi(b,A)=\varphi(b^{\prime},A)$, i.e., $b^{\prime}\vDash p_{0}$. ∎

The following theorem is the main result of this paper.

Theorem 14**.**

Fix a theory $T$ and a formula $\varphi(x,y)$. Then the following are equivalent:

1. (1)

$\varphi(x,y)$* is NIP.* 2. (2)

There is an integer $K(\varphi)$ such that for every model $M\vDash T$ and for every finite nonempty $A\subseteq M^{y}$ and $p\in S_{\varphi}(A)$, there is a formula $\psi(y,z)$ where $z=\left\langle z_{i}\,\middle|\,i<K(\varphi)\right\rangle$ which is a finite disjunction of formulas each of the form

[TABLE]

where $m\leq K(\varphi)$, $s\subseteq m$ and for every $t<m$, $s_{t}\subseteq K(\varphi)$ and $\varepsilon_{i,t}<2$ for all $i\in s_{t}$ such that $\psi(y,a)$ define $p$ for some $a\in A^{z}$. 3. (3)

There is an integer $K(\varphi)$ such that for every model $M\vDash T$ and for every finite nonempty $A\subseteq M^{y}$ and $p\in S_{\varphi}(A)$, there is a formula $\psi(y,z)$ where $z=\left\langle z_{i}\,\middle|\,i<K(\varphi)\right\rangle$ which is a finite disjunction of formulas each of the form

[TABLE]

where $m\leq K(\varphi)$, $s\subseteq m$ and for every $t<m$, $s_{t}\subseteq K(\varphi)$ and $\varepsilon_{i,t}<2$ for all $i\in s_{t}$ such that $\psi(y,a)$ define $p$ for some $a\in A^{z}$. 4. (4)

$\varphi(x,y)$* have UDTFS in $T$.*

We start with a proof of (1) implies (2), (3), so assume (1). By Remark 7 and (1), the VC-dimension of the family $\left\{\varphi(M,b)\,\middle|\,b\in M^{y}\right\}$ is bounded by some constant integer $VC(\varphi)$.

Towards a proof of (2) and (3), for the next few claims we fix $M\vDash T$, a finite $A\subseteq M^{y}$ and $p\in S_{\varphi}(A)$. Fix also $c\vDash p$ (exists in $M$ since $A$ is finite).

We will produce integers $N,J,k$ depending only on $VC(\varphi)$ and not on $M,A$ or $p$. From these we will be able to construct the defining formula $\psi$.

Definition 15**.**

For $n<\omega$, we say that $f:A^{n}\times 2^{n}\longrightarrow M$ is an $(n,\varphi)$-Skolem function if for every $(\overline{a},\overline{\varepsilon})\in A^{n}\times 2^{n}$, if there is $b\in M^{x}$ such that $M\vDash\mathord{\mathord{\bigwedge}_{i<n}}\varphi(b,a_{i})^{\varepsilon_{i}}$ then $M\vDash\mathord{\bigwedge}_{i<n}\varphi(f(\overline{a},\overline{\varepsilon}),a_{i})^{\varepsilon_{i}}$. The function $f$ is a $\varphi$-Skolem function if it is an $(n,\varphi)$-Skolem function for some $n<\omega$.

For every $\overline{a}\in A^{n}$ and $b\in M^{x}$, let $\bar{\varepsilon}^{(\bar{a},b)}\in 2^{n}$ be the unique tuple $\overline{\varepsilon}$ for which $M\vDash\mathord{\mathord{\bigwedge}_{i<n}}\varphi(b,a_{i})^{\varepsilon_{i}}$ (when $n=1$ we write $\varepsilon^{(a,b)}$).

Claim 16*.*

There is some integer $N=N(VC(\varphi))$ such that for every probability measure $\mu$ on $A$, there is a tuple $\overline{a}\in A^{\leq N}$ such that for every $(|\overline{a}|,\varphi)$-Skolem function $f$ and for every $b\in M^{x}$ we have that $\mu(\left\{a\in A\,\middle|\,\varphi(b,a)\leftrightarrow\varphi(f(\overline{a},\overline{\varepsilon}^{(\overline{a},b)}),a)\right\})\geq\frac{2}{3}$.

Proof.

Consider the set $\mathcal{S}=\left\{\varphi(c_{1},A)\mathrel{\triangle}\varphi(c_{2},A)\,\middle|\,c_{1},c_{2}\in M^{x}\right\}\subseteq\SS(A)$.

By Fact 8, the formula $\psi(xx^{\prime},y)=\varphi(x,y)\mathrel{\triangle}\varphi(x^{\prime},y)$ is NIP and has a finite VC-dimension which is moreover bounded in terms of $VC(\varphi)$. By Fact 9 it follows that the same is true for the family $\mathcal{S}$. Let $N$ be the number provided by the VC-theorem for this bound and $\epsilon=\frac{1}{3}$. It then follows that for any $\mu$ as above there is a multiset $E\subseteq A$ with $|E|\leq N$ such that $|\mu(S)-\frac{|S\cap E|}{|E|}|\leq\frac{1}{3}$ for every $S\in\mathcal{S}$. Let $\overline{a}=\left\langle a_{i}\,\middle|\,i<n\right\rangle\in A^{\leq N}$ be a tuple listing $E$. Now, fix $b\in M^{x}$ and a $\varphi$-Skolem function $f$ as above. Let $S_{0}=\varphi(b,A)\mathrel{\triangle}\varphi(f(\overline{a},\overline{\varepsilon}^{(\overline{a},b)}),A)$. Then for every $i<n$:

[TABLE]

[TABLE]

i.e., $E\cap S_{0}=\emptyset$. Therefore $|\mu(S_{0})|\leq\frac{1}{3}$ and we are done. ∎

Remark 17*.*

In fact we will only use Claim 16 with $b=c$, i.e., considering $\mathcal{S}^{\prime}=\left\{\varphi(c_{1},A)\mathrel{\triangle}\varphi(c,A)\,\middle|\,c_{1}\in M^{x}\right\}$. It is easy to see that the VC-dimension of $\mathcal{S}^{\prime}$ equals the VC-dimension of $\left\{\varphi(c_{1},A)\,\middle|\,c_{1}\in M^{x}\right\}$ (both shatter the same subsets of $A$) so we do not really require Fact 8 for the rest, but the current form of the claim is a bit more uniform.

The following claim is a translation of [MY16, Claim 3.1]. It can actually be deduced from there but for the completeness of the paper we chose to include the proof.

Claim 18*.*

There is an integer $J=J(VC(\varphi))$ such that for every tuple of $\varphi$-Skolem functions $\left\langle f_{n}\,\middle|\,n\leq N\right\rangle$ where $f_{n}$ is an $(n,\varphi)$-Skolem function, there are tuples $\overline{a_{0}},...,\overline{a_{m-1}}\in A^{\leq N}$ for $m\leq J$ such that for every $a\in A$

[TABLE]

Proof.

Let $J=J(\varphi)$ be the number we get by applying the VC-theorem on $VC(\varphi)$ and $\epsilon=\frac{1}{8}$.

Let $\left\langle f_{n}\,\middle|\,n\leq N\right\rangle$ be a sequence of $\varphi$-Skolem functions as in the claim.

By choice of $N$ from Claim 16, for every probability measure $\mu$ on $A$ there is a tuple $\overline{a}\in A^{\leq N}$ such that $\mu(\left\{a\in A\,\middle|\,\varphi(c,a)\leftrightarrow\varphi(f_{|\overline{a}|}(\overline{a},\overline{\varepsilon}^{(\overline{a},c)}),a)\right\})\geq\frac{2}{3}$.

For $\overline{a}\in A^{\leq N}$, let $h_{\overline{a}}=f_{|\overline{a}|}(\overline{a},\overline{\varepsilon}^{(\overline{a},c)})$. Let $\mathcal{H}=\left\{h_{\overline{a}}\,\middle|\,\overline{a}\in A^{\leq N}\right\}$. Enumerate $A=\left\langle a_{i}\,\middle|\,i<|A|\right\rangle$ and $\mathcal{H}=\left\langle h_{j}\,\middle|\,j<|\mathcal{H}|\right\rangle$, and define an $|A|\times|\mathcal{H}|$-matrix $B$ by setting

[TABLE]

For any probability measure $\mu$ on $A$, treat $\mu$ as a distribution vector of length $|A|$. For $j<|\mathcal{H}|$, let $v^{j}\in 2^{|\mathcal{H}|}$ be such that $v_{j^{\prime}}^{j}=1$ iff $j=j^{\prime}$ and let $\widetilde{A}_{j}=\left\{a_{i}\in A\,\middle|\,B_{i,j}=1\right\}$. We have that (when both $v^{j}$ and $\mu$ are treated as column vectors):

[TABLE]

[TABLE]

In particular we have $\underset{\mu\in\triangle^{|A|}}{\min}\underset{\nu\in\triangle^{|\mathcal{H}|}}{\max}\mu^{t}B\nu\geq\frac{2}{3}$ where (in general) $\triangle^{n}$ is the set of all probability measures on the set $n=\{0,\ldots,n-1\}$.

Fact 19**.**

[vN28]** (von Neumann’s minimax theorem) for every matrix $M\in\mathbb{R}^{n\times m}$

[TABLE]

By von Neumann’s minimax theorem it follows that $\underset{\nu\in\triangle^{|\mathcal{H}|}}{\max}\underset{\mu\in\triangle^{|A|}}{\min}\mu^{t}B\nu\geq\frac{2}{3}$. Therefore, there is some $\nu\in\triangle^{|\mathcal{H}|}$ such that for every $\mu\in\triangle^{|A|}$, $\mu^{t}B\nu\geq\frac{2}{3}$. For $i<|A|$, let $u^{i}\in 2^{|A|}$ be such that $u_{i^{\prime}}^{i}=1$ iff $i=i^{\prime}$. In particular, we have that

[TABLE]

where $\widetilde{\mathcal{H}_{i}}=\left\{h_{j}\in\mathcal{H}\,\middle|\,B_{i,j}=1\right\}$. Thus, for every $a\in A$ we have

[TABLE]

Consider the sets $S^{\varepsilon}=\left\{\varphi(\mathcal{H},a)^{\varepsilon}\,\middle|\,a\in M^{y}\right\}$ for $\varepsilon<2$. Recall that the VC-dimension of $S^{0}$ is bounded by $VC(\varphi)$. By the choice of $J$ we can apply the VC-theorem on $S^{0},\nu$ and $\epsilon=\frac{1}{8}$ and get that there is a multiset $F=\left\{h_{\overline{a_{i}}}\,\middle|\,i<m\right\}\subseteq\mathcal{H}$ such that $m\leq J$ and $\left|\nu(\varphi(\mathcal{H},a))-\frac{|F\cap\varphi(\mathcal{H},a)|}{|F|}\right|\leq\frac{1}{8}$ for every $a\in M^{y}$.

Remark 20*.*

For every probability measure $p$ on a finite set $X$ and every finite multiset $F\subseteq X$ if $\left|p(Y)-\frac{|F\cap Y|}{|F|}\right|\leq\epsilon$ for some $Y\subseteq X$ and $\epsilon>0$ then

[TABLE]

By Remark 20 it follows that for every $a\in M$

[TABLE]

Now note that for every $a\in M^{y},b\in M^{x}$ we have that $D_{a,b}\in S^{0}$ or $D_{a,b}\in S^{1}$, where $D_{a,b}=\left\{h_{\overline{a}}\in\mathcal{H}\,\middle|\,M\vDash(\varphi(b,a)\leftrightarrow\varphi(h_{\overline{a}},a))\right\}$ (if $M\vDash\varphi(b,a)$ then $D_{a,b}=\varphi(\mathcal{H},a)$ and $D_{a,b}=\neg\varphi(\mathcal{H},a)$ otherwise). So for every $D_{a,b}$ we have $\left|\nu(D_{a,b})-\frac{|F\cap D_{a,b}|}{|F|}\right|\leq\frac{1}{8}$ which implies that $\frac{|F\cap D_{a,b}|}{m}\geq\nu(D_{a,b})-\frac{1}{8}$. As $\nu(D_{a,c})\geq\frac{2}{3}$ for all $a\in A$ (by the choice of $\nu$), it then follows that for every $a\in A$ we have $\frac{|F\cap D_{a,c}|}{m}\geq\frac{2}{3}-\frac{1}{8}>\frac{1}{2}$. Thus, for all $a\in A$, $|F\cap D_{a,c}|>\frac{m}{2}$ i.e., $|\left\{t<m\,\middle|\,M\vDash(\varphi(c,a)\leftrightarrow\varphi(h_{\overline{a_{t}}},a))\right\}|>\frac{m}{2}.$ ∎

Now, let us finish the proof of (1) implies (2), (3) from Theorem 14.

Proof of (1) implies (2),(3).

By Corollary 13, there is an integer $k=k(VC(\varphi),N)$ such that for all $n\leq N$ and $(\overline{a},\overline{\varepsilon})\in A^{n}\times 2^{n}$ if $\chi_{(\overline{a},\overline{\varepsilon})}(x)=\bigwedge_{i<|\overline{a}|}\varphi(x,a_{i})^{\varepsilon_{i}}$ is consistent then there are $p_{0}^{(\overline{a},\overline{\varepsilon})}\in S_{\varphi}(A)$ and $a_{0}^{(\overline{a},\overline{\varepsilon})},...,a_{k-1}^{(\overline{a},\overline{\varepsilon})}\in A$ such that, letting $A_{0}^{(\overline{a},\overline{\varepsilon})}=\left\{a_{i}^{(\overline{a},\overline{\varepsilon})}\,\middle|\,i<k\right\}$, we have that $p_{0}^{(\overline{a},\overline{\varepsilon})}\vdash\chi_{(\overline{a},\overline{\varepsilon})}(x)$ and $p_{0}^{(\overline{a},\overline{\varepsilon})}|_{A_{0}}\vdash p_{0}^{(\overline{a},\overline{\varepsilon})}$. Now, for every such $\overline{a},\overline{\varepsilon}$, define $f_{n}^{*}(\overline{a},\overline{\varepsilon})=h$ for some $h\vDash p_{0}^{(\overline{a},\overline{\varepsilon})}.$ It follows that for all $n\leq N$, $f_{n}^{*}$ is an $(n,\varphi)$-Skolem function.

By Claim 18, for every sequence of $\varphi$-Skolem functions $\left\langle f_{n}\,\middle|\,n\leq N\right\rangle$ (such that for all $n\leq N$, $f_{n}$ is an $(n,\varphi)$-Skolem function), there is some $m\leq J$ and tuples $\overline{a_{0}},...,\overline{a_{m-1}}\in A^{\leq N}$ such that for every $a\in A$

[TABLE]

In particular this true for $\left\langle f_{n}^{*}\,\middle|\,n\leq N\right\rangle$, hence we get that there are $h_{0},...,h_{m-1}\in M^{x}$ (namely $h_{t}=f_{|\overline{a_{t}}|}^{*}(\overline{a_{t}},\overline{\varepsilon}^{(\overline{a_{t}},c)})$) such that:

• •

For every $t<m$, $\operatorname{tp}_{\varphi}(h_{t}/A)$ is $k$-isolated, i.e., there are $a_{0}^{t},...,a_{k-1}^{t}\in A$ such that for every $h^{\prime}\in M^{x}$ if for every $i<k$, $M\vDash\varphi(h_{t},a_{i}^{t})\leftrightarrow\varphi(h^{\prime},a_{i}^{t})$ then $\operatorname{tp}_{\varphi}(h_{t}/A)=\operatorname{tp}_{\varphi}(h^{\prime}/A)$, and

• •

We have that for every $a\in A$,

[TABLE]

We claim that the following formula (which is over $A$)

[TABLE]

defines the type $p$.

Indeed: first assume that $\varphi(x,a)\in p$. Then by the second bullet we have that

[TABLE]

And hence $\psi\left(a\right)$ holds.

Now suppose $M\vDash\psi(a)$ for $a\in A$. Then there are $h^{\prime}_{0},...,h^{\prime}_{m-1}$ witnessing that $\psi$ holds. But since $h^{\prime}_{t}$ agrees with $h_{t}$ on $a_{0}^{t},...,a_{k}^{t}$ for every $t<m$, it follows by the first bullet that $\operatorname{tp}_{\varphi}(h_{t}/A)=\operatorname{tp}_{\varphi}(h_{t}^{\prime}/A)$ and in particular $\operatorname{tp}_{\varphi}(h_{t}/a)=\operatorname{tp}_{\varphi}(h_{t}^{\prime}/a)$ which implies that $|\left\{t<m\,\middle|\,\varphi(h_{t},a)\right\}|>\frac{m}{2}$. Towards a contradiction, if $\varphi(x,a)\notin p$ then by the second bullet it follows that $M\vDash\neg\varphi(h_{t},a)$ for more than half the $t$’s but there are more than half the $t$’s for which $M\vDash\varphi(h_{t},a)$ which is impossible.

To see that (2) holds just note that

[TABLE]

is equivalent to

[TABLE]

This proves (2), where $K(\varphi)$ can be easily recovered from the proof using $N(VC(\varphi))$, $J(VC(\varphi))$ and $k(VC(\varphi),N(VC(\varphi))$ — a rough estimate is $K(\varphi)\leq J(VC(\varphi))\cdot k(VC(\varphi),N(VC(\varphi))$ (assuming both are positive which we may).

In order to show that (3) holds note that for every $a\in A$ and every $s\subseteq m$ we have:

[TABLE]

if and only if

[TABLE]

Indeed, this follows easily from the first bullet as above (and the fact that for all $t<m$ the formula $\bigwedge_{i<k}\varphi(x,a_{i}^{t})^{\varepsilon^{(a_{i}^{t},h_{t})}}$ is consistent).

From the above we get that for every $a\in A$ we have that $M\vDash\psi(a)\Leftrightarrow\psi^{\prime}(a)$ where

[TABLE]

which gives (3). ∎

Finally, let us finish the proof.

Proof of Theorem 14.

Assuming either (2) or (3), it is clear that there are at most finitely many formulas of the forms described there (note that the number of disjuncts in the formula is bounded). Thus by coding finitely many formulas into one as in [Gui12, Lemma 2.5] we get (4).

(4) implies (1) easily follows from type-counting argument as mentioned in the introduction. From (4) it follows that for any finite set $A\subseteq M^{y}$ for any $M\vDash T$, $|S_{\varphi}(A)|\leq|A|^{|z|}$ where $\psi(y,z)$ uniformly defines $\varphi$-types. On the other hand, if the VC-dimension of $\varphi$ is not bounded in $T$, then we can find finite sets $A$ as above such that $|S_{\varphi}(A)|$ is exponential in $|A|$, contradiction. ∎

Remark 21*.*

Note that from the proof of Theorem 14 one can extract an explicit bound for the length of the variable $z$ in the defining formula $\psi(y,z)$ that suits $\varphi(x,y)$ (as in Definition 4) which depends only on $VC(\varphi)$. An inspection of the proof can give a better description of the formula given in the formulation of Theorem 14. Any improvement on the bounds we used will automatically induce a shorter formula.

3. Open questions

3.1. NIP defining formula

Question 22**.**

Can we improve Theorem 14 as to ensure that the defining formula $\psi(y,z)$ is itself NIP?

Let us justify this question in two instances, one is when $\varphi$ is stable in $T$, and the other is when $T$ has definable Skolem functions.

3.1.1. Stable formulas

Suppose that $T$ is a theory. As in Definition 6, a formula $\varphi(x,y)$ is* stable in $T$* if it is stable in any completion of $T$ (see e.g., [TZ12, Theorem 8.2.3]; we omit “in $T$” for the rest of the discussion). If $\varphi(x,y)$ is stable then a much stronger result than UDTFS holds: there is some formula $\psi(y,z)$ such that for any $M\vDash T$ and any $A\subseteq M^{y}$, every $\varphi$-type $p\in S_{\varphi}(A)$ is definable by a formula of the form $\psi(y,a)$ for some $a\in A^{z}$. This can be deduced using the 2-rank as in [She90, Chapter II, Theorem 2.12 (3)]. However, the formula that this proof gives involves quantifiers so it is not obviously stable. Using the apparatus of non-forking extensions one can overcome this as we now explain (this is probably well-known, but it is not stated explicitly like this as far as we know).

First, for every $M\vDash T$, any $\varphi$-type over $M$ is definable over $M$ by some formula $\psi(y,m)$ where $\psi(y,z)$ is a Boolean combination of (positive) instances of $\varphi^{opp}$, see e.g., [Pil96, Lemma 2.2(i)]. In particular, $\psi(y,z)$ is itself stable (see e.g., [Pil96, Lemma 2.1]).

If $A\subseteq M\vDash T$ is any set and $p\in S_{\varphi}(A)$ (in the notation of the previous section we should have written $p\in S_{\varphi}(A^{y})$, but we ignore this for now), then $p$ has a non-forking extension $q\in S_{\varphi}(M)$ which is definable over $\operatorname{acl}^{\operatorname{eq}}(A)$ via some formula $\theta(y,a)$ (see e.g., [Pil96, Lemma 2.7]). By the first paragraph $q$ is also definable over $M$ via a Boolean combination of instances of $\varphi^{opp}$, so that $\theta(y,a)$ is itself equivalent in $M^{\operatorname{eq}}$ to a Boolean combination of instances of $\varphi^{opp}$.

Proposition 23**.**

Suppose that $M\vDash T$, $\psi(x,y)$ and $\theta(x,z)$ are formulas and $a\in M^{y},a^{\prime}\in M^{z}$. Suppose that $\psi(x,a)$ is equivalent to $\theta\left(x,a^{\prime}\right)$ in $M$ and that $\theta(x,z)$ is stable. Then there is a formula $\psi^{\prime}(x,y)$ such that $\psi^{\prime}(x,a)$ is equivalent to $\psi(x,a)$ in $M$ and $\psi^{\prime}(x,y)$ is stable.

Proof.

Let $\tau(yz)=\forall x(\psi(x,y)\leftrightarrow\theta(x,z))$ so that $M\vDash\tau(aa^{\prime})$. Let $\psi^{\prime}(x,y)=\psi(x,y)\land\exists z\tau(yz)$. Then $\psi^{\prime}(x,a)$ is equivalent to $\psi(x,a)$ in $M$ and $\psi^{\prime}$ is stable: suppose towards a contradiction that $\left\langle b_{i}a_{i}\,\middle|\,i<\omega\right\rangle$ is an indiscernible sequence which witnesses that $\psi^{\prime}$ has the order property ($\psi^{\prime}(b_{i},a_{j})$ iff $i\leq j$) in some model $N\vDash T$. Then, since e.g., $\psi^{\prime}(b_{0},a_{0})$ holds, by indiscernibility it follows that for every $j<\omega$, there exists some $a_{j}^{\prime}$ such that $\tau(a_{j}a_{j}^{\prime})$ holds. Hence we get that $i\leq j$ iff $\theta(b_{i},a_{j}^{\prime})$, contradicting the stability of $\theta$. ∎

Continuing our discussion from above, Proposition 23 implies that $q$ is definable over $\operatorname{acl}^{\operatorname{eq}}(A)$ via some formula $\theta^{\prime}(y,a)$ such that $\theta^{\prime}(y,z)$ is stable.

Let $\chi(z,c)\in\operatorname{tp}(a/A)$ be an algebraic formula of minimal (finite) size so that if $a^{\prime}\vDash\chi$ then $a^{\prime}\equiv_{A}a$. Let $\beta(y,c)=\exists z\chi(z,c)\land\theta^{\prime}(y,z)$. Then $\beta(y,c)$ is equivalent (in $M$) to $\bigvee_{a^{\prime}\vDash\chi(z,c)}\theta^{\prime}(y,a^{\prime})$ and $\bigvee_{i<m}\theta^{\prime}(y,z_{i})$ is stable as a Boolean combination of stable formulas (where $m$ is the size of $\chi(M,c)$). Hence by Proposition 23 there is some formula $\beta^{\prime}(y,w)$ such that $\beta^{\prime}(y,c)$ defines $p$ and $\beta^{\prime}(y,w)$ is stable.

Now, if we had started with $A\subseteq M^{y}$, then we could have let $A^{\prime}$ be the set of all elements appearing in some tuple from $A$, and do the same process.

From all this we got that:

Corollary 24**.**

If $\varphi(x,y)$ stable, $M\vDash T$, $A\subseteq M^{y}$ and $p\in S_{\varphi}(A)$ then $p$ is $A$-definable via a stable formula $\psi(y,z)$.

And a uniform version:

Corollary 25**.**

Suppose that $T$ is any $L$-theory and that $\varphi(x,y)$ is stable. Then there is a stable formula $\psi(y,z)$ such that for all $M\vDash T$, all $A\subseteq M^{y}$ with $|A|\geq 2$ and all $p\in S_{\varphi}(A)$ there is some $a\in A^{z}$ such that $\psi(y,a)$ defines $p$.

Proof.

By compactness, as in the proof of [She90, Chapter II, Theorem 2.12(3)].

First we show that (*) there are finitely many stable formulas which work for every $p\in S_{\varphi}(A)$. Indeed, assume not and add a new predicate $P(y)$ to the language and consider the partial type $\Gamma(x)$ in the language $L\cup\{P\}$ consisting of formulas $\theta_{\psi}(x)$ where

[TABLE]

for every stable formula $\psi(y,z)$ (where $z$ is a tuple of copies of $y$).

By our assumption towards a contradiction, for all stable formulas $\psi_{0}(y,z_{0}),\ldots,\psi_{k-1}(y,z_{k-1})$ there is some $M\vDash T$, $A\subseteq M^{y}$ and a type $p\in S_{\varphi}(A)$ which is not definable by any of the formulas $\psi_{0},\ldots,\psi_{k-1}$. This means that $\Gamma(x)$ is consistent. By compactness there is a model $M\models T$, $A\subseteq M$ and $a\in M^{x}$ such that $\operatorname{tp}_{\varphi}(a/A)$ is not definable by any stable formula $\psi(y,z)$. But this contradicts Corollary 24. This shows (*).

Now using the standard coding trick as in [Gui12, Lemma 2.5] we can code finitely many formulas $\psi_{0},\ldots,\psi_{k-1}$ into one formula $\psi$. We leave it to the reader to make sure that if all the formulas $\psi_{0},\ldots,\psi_{k-1}$ are stable, then so is $\psi$. ∎

3.1.2. Having definable Skolem functions

Proposition 26**.**

If $T$ is any theory with definable Skolem functions and $\varphi(x,y)$ is an NIP formula, then there is a formula $\psi(y,z)$ which uniformly defines $\varphi$-types over finite sets and such that $\psi(y,z)$ is itself NIP.

Proof.

By inspecting the proof of Theorem 14 (1) implies (2), one sees that in the case when $T$ has definable Skolem functions, then we would not have to use Corollary 13 at all. Instead, we could define the $\varphi$-type $p\in S_{\varphi}(A)$ by

[TABLE]

where $c\models p$ and the $\bar{a}_{t}$’s are provided by Claim 18 for the definable Skolem whose existence we assume, denoted by $f_{n,\bar{\varepsilon}}$ (i.e., $f_{n,\overline{\varepsilon}}(\overline{a})$ returns some element $b$ satisfying $\bigwedge_{i<n}\varphi(x,a_{i})^{\varepsilon_{i}}$ if such exists). In other words, the formula $\psi(y,b)$ defining $p$ is a Boolean combination of instances of formulas of the form $\varphi(f(b),y)$ for $b$ a tuple from $A$ and $f$ some definable function.

In general, when $\varphi(x,y)$ is an NIP formula and $f$ is some definable function, then $\psi(z,y)=\varphi(f(z),y)$ is also NIP (if $\left\{\psi(M,b)\,\middle|\,b\in M^{y}\right\}$ shatters some set $A\subseteq M^{z}$ then $\left\{\varphi(M,b)\,\middle|\,b\in M^{y}\right\}$ shatters the image of $A$ under $f$ which has the same size as $A$). This means (by Fact 8) that $\psi(y,z)$ is NIP.

As in the stable case, coding finitely many NIP formulas into one gives an NIP formula, so we are done. ∎

3.2. Honest definitions

Definition 27**.**

[Sim15, Definition 3.16 and Remark 3.14] Work in some theory $T$. Suppose that $\varphi(x,y)$ is a formula, $A\subseteq M^{y}$ some set and $p\in S_{\varphi}(A)$. Say that a formula $\psi(y,z)$ (over $\emptyset$) is an honest definition of $p$ if for every finite $A_{0}\subseteq A$ there is some $b\in A^{z}$ such that for all $a\in A$, if $\psi(a,b)$ holds then $\varphi(x,a)\in p$ and for all $a\in A_{0}$ the other direction holds: if $\varphi(x,a)\in p$ then $\psi(a,b)$ holds.

It is proved in [Sim15, Theorem 6.16], [CS15, Theorem 11] that if $T$ is NIP then for every $\varphi(x,y)$ there is a formula $\psi(y,z)$ that serves as an honest definition for any type in $S_{\varphi}(A)$ provided that $|A|\geq 2$.

Question 28**.**

Is this true only assuming that $\varphi(x,y)$ is NIP?

Note that for the proof in [Sim15, Theorem 6.16], [CS15, Theorem 11], only a very mild assumption of NIP is actually needed. See [CS15, Remark 16].