#  Remarks on convergence of Morley sequences

Karim Khanaki
Arak University of Technology

Dedicated to the memory of my first teacher Fatemeh Mardani Partially
supported by IPM grant 1400030118

###### Abstract

We refine results of Gannon [G21, Theorem 4.7] and Simon [S15a, Lemma 2.8] on
convergence of Morley sequences. We then introduce the notion of eventual
$NIP$, as a property of a model, and prove a variant of [KP18, Corollary 2.2].
Finally, we give new characterizations of generically stable types (for
countable theories) and reinforce the main result of Pillay [P18] on the
model-theoretic meaning of Grothendieck’s double limit theorem.

##  1 Introduction

Pillay and Tanović [PT11] introduced the notion of generically stable type,
for arbitrary theories, as an abstraction of the crucial properties of
definable types in stable theories. All invariant types in $NIP$ theories and
all generically stable types in arbitrary theories share an important
phenomenon: convergence of Morley sequences. Using this phenomenon/property,
although it is not explicitly mentioned, Simon [S15a] proved the following
interesting result:

Simon’s Lemma111[S15a, Lemma 2.8]. In this article, when we refer to Simon’s
Lemma, we mean this result.: Let $T$ be a countable $NIP$ theory and $M$ a
countable model of $T$. Suppose that $p(x)\in S(\cal U)$ is finitely
satisfiable in $M$. Then there is a sequence $(c_{i})$ in $M$ such that $\lim
tp(c_{i}/{\cal U})=p$.

The present paper aims to focus on convergence of Morley sequences. The core
of our observations/proofs here is that the convergence of tuples/types
depends on a certain type of formulas, namely symmetric formulas. We show that
a sequence of types converges if and only if there are some symmetric formulas
that are not true in the sequence.

On the other hand, the origin of Simon’s lemma is related to the following
crucial theorem in functional analysis due to Bourgain, Fremlin, and Talagrand
[BFT78, Thm. 3F]:

BFT Theorem: Let $X$ be a Polish space. Then the space $B_{1}(X)$ of all Baire
1 functions on $X$ is an angelic space with the topology of pointwise
convergence.

In [Kha21a, Appendix A], it is shown that complete types (not just
$\phi$-types) can be coded by suitable functions, and a refinement of Simon’s
lemma is given using the BFT theorem. Recall that every point in the closure
of a relatively compact set of an angelic space is the limit of a sequence of
its points, and relatively compact sets of $B_{1}(X)$ possess a property
similar to $NIP$. (Cf. [KP18].) In this paper we aim to give a model theoretic
version of the Bourgain-Fremlin-Talagrand result in the terms of convergent
Morley sequences.222Although we only use one analytical/combinatorial result
(Fact 2.10), we will explain that it is not even needed and that all arguments
in this paper are model-theoretic. Cf. Remark 2.12(iv) below. More precisely,
we prove that:

Theorem A: Let $T$ be a countable theory and $M$ a countable model of
$T$.333We can consider countable fragments of (uncountable) theories, however,
to make the proofs more readable, we assume that the theory is countable.
(1) Suppose that $p(x)\in S(\cal U)$ is finitely satisfiable in $M$ and there
is a Morley sequence $(d_{i})$ of $p$ over $M$ such that $(tp(d_{i}/{\cal
U}):i<\omega)$ converges. Then there is a sequence $(c_{i})\in M$ such that
$\lim tp(c_{i}/{\cal U})=p$.
(2) Furthermore, the following are equivalent:
(i) $M$ is eventually $NIP$ (as in Definition 3.1).
(ii) For any $p(x)\in S(\cal U)$ which is finitely satisfiable in $M$, there
is a sequence $(c_{i})\in M$ such that the sequence $(tp(c_{i}/{\cal
U}):i<\omega)$ $DBSC$-converges to $p$ (as in Definition 3.4).
(iii) For any $p(x)\in S(\cal U)$ which is finitely satisfiable in $M$, there
is a Morley sequence $(d_{i})$ of $p$ over $M$ such that $(tp(d_{i}/{\cal
U}):i<\omega)$ converges.

Recall that Morley sequences in $NIP$ theories are convergent (cf. Definition
2.9).444This is a consequence of indiscernibility of Morley sequences and
countability of theory. If not, one can find a Morley sequence $(a_{i})$ and a
formula $\phi(x)$ such that $\models\phi(a_{i})$ iff $i$ is even, a
contradiction. Therefore, as the theory $T$ in Theorem A is arbitrary, the
equivalences (i)–(iii) of Theorem A(2) refine Simon’s lemma above. On the
other hand, a result of Gannon [G21, Thm. 4.7] asserts that:

Gannon’s theorem: Let $T$ be a countable theory and $M$ a (not necessarily
countable) model of $T$. Suppose that $p(x)\in S(\cal U)$ is generically
stable over $M$. Then there is a sequence $(c_{i})$ in $M$ such that $\lim
tp(c_{i}/{\cal U})=p$.

This follows from Theorem A(1) and the fact that every generically stable type
over $M$ is generically stable over a countable elementary substructure of
$M$. We also give a refinement of Gannon’s theorem. In fact, we give a new
characterization of generically stable types for countable theories:

Theorem B: Let $T$ be a countable theory, $M$ a model of $T$, and $p(x)\in
S({\cal U})$ a global $M$-invariant type. The following are equivalent:
(i) $p$ is generically stable over $M$.
(ii) $p$ is definable over $M$, AND there is a sequence $(c_{i})$ in $M$ such
that $(tp(c_{i}/{\cal U}):i<\omega)$ $DBSC$-converges to $p$ (as in Definition
3.4).
Suppose moreover that $T$ has $NSOP$, then (iii) below is also equivalent to
(i), (ii) above:
(iii) there is a sequence $(c_{i})$ in $M$ such that $(tp(c_{i}/{\cal
U}):i<\omega)$ $DBSC$-converges to $p$.

Notice that, as $DBSC$-convergence is strictly stronger than usual
convergence, Theorem B is a clear refinement of Gannon’s theorem. Moreover,
this result can be lead to an answer to [G21, Question 4.15].

Theorems A and B allow us to reinforce the main result of [P18] on generic
stability in a model. That is,

Theorem C: Let $T$ be a (countable or uncountable) theory, and $M$ be a model
of $T$. The following are equivalent:
(i) Any type $p\in S_{x}(M)$ has an extension to a global type $p^{\prime}\in
S_{x}({\cal U})$ which is generically stable over $M$.
(ii) $M$ has no order (as in Definition 4.8) AND $M$ is eventually $NIP$.

It is worth mentioning that Gannon’s theorem based on the idea of Simon’s
lemma, and our results/observations are based on ideas of both of them. This
paper is a kind of companion-piece to [KP18] and [Kha17], although here we are
mainly concerned with model-theoretic proofs of variants of results from
[KP18].

This paper is organized as follows. In Section 2, we fix some model theoretic
conventions. We will also prove Theorem A(1). (Cf. Theorem 2.11.) In Section
3, we will provide all necessary functional analysis notions, and introdce the
notion of eventual $NIP$. We will also prove Theorem A(2). (Cf. Theorem 3.6.)
In Section 4, we will study generically stable types in arbitrary/countable
theories. We will also prove Theorem B and Theorem C. (Cf. Theorems 4.4,
4.10.) At the end paper we conclude some remarks/questions on future
generalizations and applications of the results/observations.

##  2 Convergent Morley sequences

The notation is standard, and a text such as [S15] will be sufficient
background. We fix a first order language $L$, a complete countable $L$-theory
$T$ (not necessarily $NIP$), and a countable model $M$ of $T$. The monster
model is denoted by $\cal U$ and the space of global types in the variable $x$
is denoted by $S_{x}({\cal U})$ or $S({\cal U})$.

######  Convention 2.1.

In this paper, when we say that $(a_{i})\subset{\cal U}$ is a sequence, we
mean the usual notion in the sense of analysis. That is, every sequence is
indexed by $\omega$. Similarly, we consider Morley sequences indexed by
$\omega$.

######  Convention 2.2.

In this paper, a variable $x$ is a tuple of length $n$ (for
$n<\omega$).555Although all arguments are true for infinite variables, to make
the proofs readable, we consider finite tuples. Sometimes we write $\bar{x}$
or $x_{1},\ldots,x_{n}$ instead of $x$. All types are $n$-type (for
$n<\omega$) unless explicitly stated otherwise. Similarly, a sequence
$(a_{i})\subset{\cal U}$ is a sequence of tuples of length $n$ (for
$n<\omega$).

######  Convention 2.3.

In this paper, when we say that $\phi$ is a formula, we mean a formula over
$\emptyset$. Otherwise, we explicitly say that $\phi$ is an $L(A)$-formula for
some set/model $A$. Although the structure of some important definitions and
proofs does not depend on the parameter at all.

######  Definition 2.4.

Let $A\subset\cal U$ and $\phi(x_{1},\ldots,x_{n})\in L(A)$. We say that
$\phi(x_{1},\ldots,x_{n})$ is symmetric if for any permutation $\sigma$ of
$\\{1,\ldots,n\\}$,

$\models\forall\bar{x}\big{(}\phi(x_{1},\ldots,x_{n})\leftrightarrow\phi(x_{\sigma{(1)}},\ldots,x_{\sigma{(n)}})\big{)}.$

For a formula $\phi(x)$ (with or without parameters) and a sequence $(a_{i})$
of $x$-tuples in $\cal U$, we write $\lim_{i\to\infty}\phi(a_{i})=1$ if there
is a natural number $n$ such that ${\cal U}\models\phi(a_{i})$ for all $i\geq
n$. If $\lim_{i\to\infty}\neg\phi(a_{i})=1$ we write
$\lim_{i\to\infty}\phi(a_{i})=0$.

For a formula $\phi(x_{i_{0}},\ldots,x_{i_{k}})$ and a sequence
$(b_{i})\in\cal U$, if there exists an $n_{\phi}$ such that for any
$i_{k}>\cdots>i_{0}>n_{\phi}$ we have ${\cal
U}\models\phi(b_{i_{0}},\ldots,b_{i_{k}})$, we write
$\lim_{i_{0}<\cdots<i_{k},\
i_{0}\to\infty}\phi(b_{i_{0}},\ldots,b_{i_{k}})=1$.

######  Definition 2.5.

(i) Let $(b_{i})$ be a sequence of elements in $\cal U$ and $A\subset\cal U$ a
set. The eventual Ehrenfeucht-Mostovski type666The $EEM$-type is defined in
[G21, Def. 4.3]. It was extracted from the notion of eventual indiscernible
sequence in [S15a]. (abbreviated $EEM$-type) of $(b_{i})$ over $A$, which is
denoted by $EEM((b_{i})/A)$, is the following (partial) type in
$S_{\omega}(A)$:

$\phi(x_{0},\ldots,x_{k})\in EEM((b_{i})/A)\iff\lim_{i_{0}<\cdots<i_{k},\
i_{0}\to\infty}\phi(b_{i_{0}},\ldots,b_{i_{k}})=1.$

(ii) Let $(b_{i})$ be a sequence of $\cal U$ and $A\subset\cal U$ a set. The
symmetric eventual Ehrenfeucht-Mostovski type (abbreviated $SEEM$-type) of
$(b_{i})$ over $A$, which is denoted by $SEEM((b_{i})/A)$, is the following
partial type in $S_{\omega}(A)$:

$\big{\\{}\phi=\phi(x_{0},\ldots,x_{n}):\phi\in EEM((b_{i})/A)\text{ and
}\phi\text{ is symmetric}\big{\\}}.$

Whenever $(b_{i})$ is $A$-indiscernible, we sometimes write $SEM((b_{i})/A)$
instead of $SEEM((b_{i})/A)$.
(iii) Let $p(x)$ be a type in $S_{\omega}(A)$ (or $S_{\omega}({\cal U})$). The
symmetric type of $p$, denoted by Sym$(p)$, is the following partial type:

$\big{\\{}\phi(x)\in p:\phi\text{ is symmetric}\big{\\}}.$

The sequence $(b_{i})$ is called eventually indiscernible over $A$ if
$EEM((b_{i})/A)$ is a complete type. In this case, for any $L(A)$-formula
$\phi(x)$, the limit $\lim_{i\to\infty}\phi(b_{i})$ is well-defined.

######  Fact 2.6 ([G21], Fact 4.2).

Let $(b_{i})$ be a sequence of elements in $\cal U$ and $A\subset\cal U$ such
that $|A|=\aleph_{0}$. Then there exists a subsequence $(c_{i})$ of $(b_{i})$
such that $(c_{i})$ is eventually indiscernible over $A$.

###### Proof.

A generalization of this observation (for continuous logic) is proved in
Proposition 5.3 of [G21]. ∎

Let $A$ be a set/model and $p(x)$ a global $A$-invariant type. The Morley type
(or sequence) of $p(x)$ is denoted by $p^{(\omega)}$. (Cf. [S15], subsection
2.2.1.) The restriction of $p(x)$ to $A$ is denoted by $p|_{A}$. A realisation
$(d_{i}:i<\omega)$ of $p^{(\omega)}|_{A}$ is called a Morley sequence of/in
$p$ over $A$.

######  Lemma 2.7.

Let $p(x)\in S(\cal U)$ be invariant over $A$, and $I=(d_{i})$ a Morley
sequence in $p$ over $A$.
(i) If there is a sequence $(c_{i})\in A$ such that $\lim
tp(c_{i}/AI)=p|_{AI}$ and $(c_{i})$ is eventually indiscernible over $AI$,
then $SEEM((c_{i})/A)=\text{Sym}(p^{(\omega)}|_{A})$.
(ii) If $p$ is finitely satisfaible in $A$ and $|A|=\aleph_{0}$, then there is
a sequence $(c_{i})$ in $A$ such that $\lim tp(c_{i}/AI)=p|_{AI}$ and
$(c_{i})$ is eventually indiscernible over $AI$. Therefore,
$SEEM((c_{i})/A)=\text{Sym}(p^{(\omega)}|_{A})$.

###### Proof.

(i): Suppose that there is a sequence $(c_{i})$ such that $\lim
tp(c_{i}/AI)=p|_{AI}$ and $(c_{i})$ is eventually indiscernible over $AI$
$(\dagger)$. Set $J=(c_{i})$.

We show that $SEEM((c_{i})/A)=SEM((d_{i})/A)=\text{Sym}(p^{(\omega)}|_{A})$.
We remind the reader that $SEM((d_{i})/A)=\text{Sym}(p^{(\omega)}|_{A})$
follows from the definition of a Morley sequence. The proof is by induction on
symmetric formulas. The base case works. Indeed, for any $L(A)$-formula
$\phi(x_{0})$, $\phi(x_{0})\in
SEEM((c_{i})/A)\iff\lim\phi(c_{i})=1\iff\phi(x_{0})\in p\iff\phi(x_{0})\in
SEM((d_{i})/A)$.

The induction hypothesis is that for any symmetric formula
$\phi(x_{0},\ldots,x_{k-1})$ in $L(A)$, $\phi(x_{0},\ldots,x_{k-1})\in
SEEM((c_{i})/A)$ if and only if $\phi(x_{0},\ldots,x_{k-1})\in
SEM((d_{i})/A)$.

Let $\phi(x_{0},\ldots,x_{k})$ be a symmetric $L(A)$-formula $(\ddagger)$.
Clearly, for any $c\in A$, the $L(A)$-formula $\phi(c,x_{1},\ldots,x_{k})$ is
symmetric. Therefore, by the induction hypothesis,

$\lim_{i\to\infty}\phi(c,c_{i+1},\ldots,c_{i+k})=\phi(c,d_{1},\ldots,d_{k})$
$(*)$.

On the other hand, since $\lim_{n\to\infty}tp(c_{n}/AI)=p|_{AI}$, we have

$\lim_{n\to\infty}\phi(c_{n},d_{1},\ldots,d_{k})=\phi(d_{k+1},d_{1},\ldots,d_{k})$
$(**)$.

To summarize, for large $n$,

$\displaystyle\lim_{j\to\infty}\phi(c_{j},c_{j+1},\ldots,c_{j+k})$
$\displaystyle\stackrel{{\scriptstyle(\dagger)}}{{=}}\lim_{n\to\infty}\lim_{i\to\infty}\phi(c_{n},c_{i+1},\ldots,c_{i+k})$
$\displaystyle\stackrel{{\scriptstyle(*)}}{{=}}\lim_{n\to\infty}\phi(c_{n},d_{1},\ldots,d_{k})$
$\displaystyle\stackrel{{\scriptstyle(**)}}{{=}}\phi(d_{k+1},d_{1},\ldots,d_{k})$
$\displaystyle\stackrel{{\scriptstyle(\ddagger)}}{{=}}\phi(d_{1},\ldots,d_{k+1}).$

This means that $\phi(\bar{x})\in SEEM(J/A)$ iff $\phi(\bar{x})\in SEM(I/A)$.

(ii): Let $I^{\prime}$ be a Morley sequence in $p$ over $A$. Since $T$ and $A$
are countable, and $p$ is finitely satisfiable in $A$, there is a sequence
$(c_{i})$ in $A$ such that $\lim tp(c_{i}/AI^{\prime})=p|_{AI^{\prime}}$.
(Notice that the closure of $\\{tp(a/AI^{\prime}):a\in A\\}\subset
S_{x}(AI^{\prime})$ is second-countable and compact,777Recall that a compact
Hausdorff space is metrizable if and only if it is second-countable. and so
metrizable. Therefore, there is a sequence $(c_{i})\in A$ such that $\lim
tp(c_{i}/AI^{\prime})=p|_{AI^{\prime}}$.)888Another argument that is more
model-theoretic is given in the first paragraph of the proof of [G21, Lemma
4.6].

By Fact 2.6, we can assume that $(c_{i})$ is eventually indiscernible over
$AI$ $(\dagger)$. That is, the type $EEM((c_{i})/AI)$ is complete. (Notice
that, as $AI$ and $T$ are countable, using Ramsey’s theorem and a diagonal
argument, there is a subsequence of $(c_{i})$ which is eventually
indiscernible over $AI$.)

By (i), $SEEM((c_{i})/A)=\text{Sym}(p^{(\omega)}|_{A})$. ∎

######  Remark 2.8.

The proof of Lemma 2.7 is essentially the same as [G21, Lemma 4.5]. The
difference is that we don’t need all formulas, but only symmetric ones. Notice
that it is not necessary to assume that $A$ is a model. It is worth recalling
that Gannon’s result is based on an idea of Simon [S15a, Lemma 2.8].

Lemma 2.7 above discusses converging of tuples, although in the rest of paper,
converging of tuples means convergence of types/tuples over the monster model,
but not small sets/models:

######  Definition 2.9.

We say that a sequence $(d_{i})\in\cal U$ of $x$-tuples converges (or is
convergent) if the sequence $(tp(d_{i}/{\cal U}):i<\omega)$ converges in the
logic topology. Equivalently, for any $L({\cal U})$-formula $\phi(x)$, the
truth value of $(\phi(d_{i}):i<\omega)$ is eventually constant. If
$(tp(d_{i}/{\cal U}):i<\omega)$ converges to a type $p$, then we write $\lim
tp(d_{i}/{\cal U})=p$ or $tp(d_{i}/{\cal U})\to p$. Notice that
$tp(d_{i}/{\cal U})\to p$ iff for any $L({\cal U})$-formula $\phi(x)$,

$\lim\phi(d_{i})=1\iff\phi(x)\in p.$

######  Fact 2.10.

Let $(d_{i})$ be a sequence in $\cal U$ of $x$-tuples. Then the following are
equivalent:
(i) $(d_{i})$ has a subsequence with no convergent subsequence.
(ii) There are a subsequence $(d_{i}^{\prime})\subseteq(d_{i})$ and a formula
$\phi(x,y)$ (with or without parameters) such that for all (finite) disjoint
subsets $E,F\subseteq\mathbb{N}$,

$\models\exists y\Big{(}\bigwedge_{i\in
E}\phi(d_{i}^{\prime},y)\wedge\bigwedge_{i\notin
F}\neg\phi(d_{i}^{\prime},y)\Big{)}.$

Furthermore, suppose that $(d_{i})$ is indiscernible. Then each of (i), (ii)
above is also equivalent to (iii) below:
(iii) The condition (ii) holds for any subsequence of $(d_{i})$. More
precisely, there is a formula $\phi(x,y)$ (with or without parameters) such
that for any subsequence $(d_{i}^{\prime})\subseteq(d_{i})$ and for all
(finite) disjoint subsets $E,F\subseteq\mathbb{N}$,

$\models\exists y\Big{(}\bigwedge_{i\in
E}\phi(d_{i}^{\prime},y)\wedge\bigwedge_{i\notin
F}\neg\phi(d_{i}^{\prime},y)\Big{)}.$

###### Proof.

The direction (i) $\Rightarrow$ (ii) follows from one of the prettiest result
in the Banach space theory due to Rosenthal, Theorem 1 in [Ros74]. (See also
Lemma 3.12 of [Kha20] or Appendix B in [Kha21].) Indeed, as $T$ is countable,
we can assume999If not, using a diagonal argument, we can find a convergent
subsequence of any subsequence. that there is a subsequence
$(c_{i})\subseteq(d_{i})$ and a formula $\phi(x,y)$ such that the sequence
$(\phi(c_{i},y):i<\omega)$ has a subsequence with no convergent subsequence.
Now use Rosenthal’s theorem for it. (On the other hand, notice that, as $T$ is
countable, every complete type can be coded by a function on a suitable space
(cf. [Kha21a, Appendix A]). This leads to an alternative argument.)
The direction (ii) $\Rightarrow$ (iii) follows from indiscernibility.
(iii) $\Rightarrow$ (ii) $\Rightarrow$ (i) are evident. ∎

We emphasize that, in Fact 2.10, the direction (i) $\Longrightarrow$ (ii)
needs countability of the theory. On the other hand, it is easy to verify
that, this fact holds for real-valued functions (or types in continuous
logic).

######  Theorem 2.11.

Let $T$ be a countable theory, $M$ a countable model,101010Tanović pointed out
to us that it is not necessary to assume that $M$ is a model. Cf. Remark
2.12(iv). and $p(x)\in S(\cal U)$ a global type which is finitely satisfiable
in $M$. Let $(d_{i})$ be a Morley sequence of $p$ over $M$. If $(d_{i})$
converges then there is a sequence $(c_{i})$ in $M$ such that $tp(c_{i}/{\cal
U})\to p$.

###### Proof.

By Lemma 2.7, we can assume that there is a sequence $(c_{i})$ in $M$ such
that $tp(c_{i}/M\cup(d_{i}))\to p|_{M\cup(d_{i})}$ and
$SEEM((c_{i})/M)=\text{Sym}(p^{(\omega)}|_{M})$. We show that $tp(c_{i}/{\cal
U})\to p$. Let $q$ be an accumulation point of $\\{tp(c_{i}/{\cal
U}):i\in\omega\\}$. Then $q|_{M\cup(d_{i})}=p|_{M\cup(d_{i})}$. Notice that,
as $q$ is finitely satisfiable in $M$ (and so $M$-invariant), the type
$q^{(\omega)}$ is well-defined.

Claim 0: $p^{(\omega)}|_{M}=q^{(\omega)}|_{M}$.

_Proof_ : The proof is by induction. The base case is
$q|_{M\cup(d_{i})}=p|_{M\cup(d_{i})}$. The induction hypothesis is that
$p^{(n+1)}|_{M}=q^{(n+1)}|_{M}$. Let $\phi(x_{n+1},x_{n},\ldots,x_{0})\in
L(M)$, and suppose that $p_{x_{n+1}}\otimes
p^{(n+1)}_{\bar{x}}\vdash\phi(x_{n+1},{\bar{x}})$, where
${\bar{x}}=(x_{n},\ldots,x_{0})$. Since $(d_{i})$ is a Morley sequence in $p$
over $M$, $\models\phi(d_{n+1},{\bar{d}})$ where
${\bar{d}}=(d_{n},\ldots,d_{0})$. By definition of Morley sequence,
$p_{x_{n+1}}\vdash\phi(x_{n+1},{\bar{d}})$ and ${\bar{d}}\models
p^{(n+1)}_{\bar{x}}|_{M}$. By the hypothesis of induction, ${\bar{d}}\models
q^{(n+1)}_{\bar{x}}|_{M}$. By the base case,
$q_{x_{n+1}}\vdash\phi(x_{n+1},{\bar{d}})$, and so by definition,
$q_{x_{n+1}}\otimes q^{(n+1)}_{\bar{x}}\vdash\phi(x_{n+1},{\bar{x}})$.
$\dashv_{\text{claim 0}}$

Claim 1: $p=q$.

_Proof_ : If not, assume for a contradiction that $p\vdash\phi(x,b)$ and
$q\vdash\neg\phi(x,b)$ for some $b\in\cal U$ and formula $\phi(x,y)$ (without
parameters). We inductively build a sequence $(a_{i})$ as follows:

$\bullet$ if $i$ is even, $a_{i}\models
p|_{M\cup\\{a_{1},\ldots,a_{i-1},b\\}}$;

$\bullet$ if $i$ is odd, $a_{i}\models
q|_{M\cup\\{a_{1},\ldots,a_{i-1},b\\}}$.

As $p^{(\omega)}|_{M}=q^{(\omega)}|_{M}$, the sequence $(a_{i})$ is
indiscernible and its type over $M$ is $p^{(\omega)}|_{M}$. Moreover,
$\phi(a_{i},b)\iff i\text{ is even}$.

As $(a_{i})$ is indiscernible, using the backward direction of [S15, Lemma
2.7], ${\cal U}\models\theta_{n,\phi}(a_{1},\ldots,a_{n})$111111According to
that argument of Claim 0, sometimes it is better to ‘reverse’ the definition
of EEM type: replace each $\phi(x_{1},\ldots,x_{n})$ by
$\phi(x_{n},\ldots,x_{1})$. This was suggested to us by Tanović. Although we
still continue the previous arrangement. where

$\theta_{n,\phi}(x_{1},\ldots,x_{n})=\forall F\subseteq\\{1,\ldots,n\\}\exists
y_{F}\Big{(}\bigwedge_{i\in F}\phi(x_{i},y_{F})\wedge\bigwedge_{i\notin
F}\neg\phi(x_{i},y_{F})\Big{)}.$

(Notice that $\theta_{n,\phi}$ is symmetric, and so
$\theta_{n,\phi}\in\text{Sym}(p^{(\omega)}|_{M})$.) This means that
$\models\theta_{n,\phi}(d_{1},\ldots,d_{n})$ for all $n$. Therefore, for any
infinite subset $I\subseteq\mathbb{N}$, the set

$\Sigma_{I}^{\phi}(y)=\Big{\\{}\bigwedge_{i\in
I\cap\\{1,\ldots,n\\}}\phi(d_{i},y)\wedge\bigwedge_{i\notin
I\cap\\{1,\ldots,n\\}}\neg\phi(d_{i},y):n\in{\mathbb{N}}\Big{\\}}$

is a partial type. This means that the sequence $\phi(d_{i},y)$ does not
converge (and even it has no convergent subsequence), a contradiction.
(Alternatively, as both $(a_{i}),(d_{i})$ are Morley sequence over $M$, there
is an automorphism $\sigma\in Aut({\cal U},M)$ which maps $a_{i}$ to $d_{i}$.
Then $\phi(d_{i},\sigma(b))$ converges iff $\phi(a_{i},b)$
converges.121212This was suggested to us by Tanović and the referee,
independently.) $\dashv_{\text{claim 1}}$

Claim 2: The sequence $(tp(c_{i}/{\cal U}):i<\omega)$ converges.

_Proof_ : If not, without loss of generality we can assume that it has no
convergent subsequence. (If $(c_{i})$ has a convergent subsequence, we can
just choose it to be our sequence and explain why it converges to $p$.) As $T$
is countable, there is a formula $\psi(x,y)$ (with or without parameters) such
that, the sequence $(\psi(c_{i},y):i<\omega)$ has no convergent
subsequence.131313If not, using a diagonal argument one can show the sequence
$(tp(c_{i}/{\cal U}):i<\omega)$ has a convergent subsequence. Then, by Fact
2.10, for any infinite subset $I\subseteq\mathbb{N}$, the set

$\Sigma_{I}^{\psi}(y)=\Big{\\{}\bigwedge_{i\in
I\cap\\{1,\ldots,n\\}}\psi(c_{i},y)\wedge\bigwedge_{i\notin
I\cap\\{1,\ldots,n\\}}\neg\psi(c_{i},y):n\in{\mathbb{N}}\Big{\\}}$

is a partial type. As $\theta_{n,\psi}$ is symmetric, this means that
$\theta_{n,\psi}\in SEEM((c_{i})/M)$, and so
$\models\theta_{n,\psi}(d_{1},\ldots,d_{n})$ for all $n$. Equivalently,
$(\psi(d_{i},y):i<\omega)$ does not converge, a contradiction.
$\dashv_{\text{claim 2}}$

Since $(tp(c_{i})/{\cal U}:i\in\omega)$ converges to say a type $r$, this type
is in the topological closure of $\\{tp(c_{i}/{\cal U}):i\in\omega\\}$. Hence
by the first portion of the argument, $r=p$. ∎

######  Remark 2.12.

(i) Let $T$ be a (countable or uncountable) theory, $M$ a model, and $p(x)\in
S(\cal U)$ a global type. The argument of Claim 1 in the proof of Theorem 2.11
shows that if there exists a Morley sequence of $p$ which is convergent, then
any Morley sequence of $p$ is convergent.
(ii) Let $T$ be a (countable or uncountable) theory, $M$ a model, and
$p(x),q(x)$ two global $M$-invariant types. If the Morley sequence of $p$ is
convergent and $p^{(\omega)}|_{M}=q^{(\omega)}|_{M}$, then the argument of
Claim 1 in the proof of Theorem 2.11 shows that $p=q$.141414Notice that we can
assume $p^{(\omega)}|_{\emptyset}=q^{(\omega)}|_{\emptyset}$ because the
formula $\phi(x,y)$ in Claim 1 has no parameters. As $T$ is arbitrary, this is
a generalization of Proposition 2.36 of [S15]. (See also Lemma 2.5 of [HP11].)
(iii) There is a converse to Theorem 2.11: Let $T$ be a countable theory, $M$
a countable model, and $p(x)\in S(\cal U)$ a global type which is finitely
satisfiable in $M$. If there is a sequence $(c_{i})$ in $M$ such that
$tp(c_{i}/{\cal U})$ $DBSC$-converges to $p$ (as in Definition 3.4), then
some/any Morley sequence of $p$ (over $M$) is convergent. (See the argument of
the direction (ii) $\Longrightarrow$ (iii) of Theorem 3.6.)
(iv) (Tanović) Proof of Claim 2 in Theorem 2.11: If not, there are formulas
$\psi(x,y)$ (without parameters) and $b\in\cal U$ such that both sets
$C_{1}=\\{c_{i}:\models\psi(c_{i},b)\\}$ and
$C_{2}=\\{c_{i}:\models\neg\psi(c_{i},b)\\}$ are infinite. Let $p_{1},p_{2}$
be accumulation points of $\\{tp(c_{i}/{\cal U}):c_{i}\in C_{1}\\}$ and
$\\{tp(c_{i}/{\cal U}):c_{i}\in C_{2}\\}$, respectively. Notice that
$p_{1}|_{(d_{i})}=p_{2}|_{(d_{i})}$. We inductively build a sequence $(a_{i})$
as follows:

$\bullet$ if $i$ is even, $a_{i}\models p_{1}|_{\\{a_{1},\ldots,a_{i-1}\\}}$;

$\bullet$ if $i$ is odd, $a_{i}\models p_{2}|_{\\{a_{1},\ldots,a_{i-1}\\}}$.

Similar to Claim 0, we have
$p_{1}^{(\omega)}|_{(d_{i})}=p_{2}^{(\omega)}|_{(d_{i})}$, and similar to
Claim 1, as the Morley sequence $(d_{i})$ is convergent, we have
$p_{1}=p_{2}$. This is a contradiction, as $p_{1}\vdash\psi(x,b)$ and
$p_{1}\vdash\neg\psi(x,b)$. (In fact, it is not necessary to assume that $M$
is a model, and we can assume that $M=(c_{i})$. However, countability remains
a key assumption.)

Although with Remark 2.12(iv), we don’t need Fact 2.10, but for better
intuition and providing basic concepts in the rest of the article, the
approach of Fact 2.10 is useful (cf. Definition 3.1 below).

######  Corollary 2.13 ([G21], Theorem 4.8.).

Let $T$ be a countable theory, $p(x)\in S(\cal U)$ and $N$ a (not necessarily
countable) model. If $p$ is generically stable over $N$, then there is a
sequence $(c_{i})\in N$ such that $tp(c_{i}/{\cal U})\to p$.

###### Proof.

As $T$ is countable, there is a countable elementary substructure $M$ of $N$
such that $p$ is generically stable over $M$, and so $p$ is finitely
satisfiable in $M$ and every Morley sequence of $p$ is convergent. (See also
Fact 4.1 below.) Then, by Theorem 2.11, there is a sequence $(c_{i})\in M$
such that $tp(c_{i}/{\cal U})\to p$. ∎

Notice that in the proof of Theorem 2.11, for any formula $\psi$ there is a
natural number $n$ such that $\theta_{n,\psi}\notin SEEM((c_{i})/M)$. This is
equivalent to a stronger version of convergence that was studied in [Kha19b]
and we will recall it in the next section. This implies that our result is
strictly stronger than Gannon’s theorem. Cf. Theorem 4.4, the direction (i)
$\Rightarrow$ (ii). This is also related to Question 4.15 of [G21].

##  3 Eventual $NIP$

In this section, we want to give a characterization of convergent Morley
sequences over countable models. First we introduce the following notions.

######  Definition 3.1.

Let $T$ be a theory, $M$ a model of it and $\phi(x,y)$ a formula (with or
without parameters).
(i) We say that $\phi(x,y)$ is eventually $NIP$ in $M$ if for any infinite
sequence $(c_{i})\in M$ there are a subsequence $(a_{i})\subseteq(c_{i})$, a
natural number $n=n_{(a_{i})}$ and subset $E\subseteq\\{1,\ldots,n\\}$ such
that for any $i_{1}<\cdots<i_{n}<\omega$, ${\cal
U}\models\psi_{\phi}(a_{i_{1}},\ldots,a_{i_{n}})$ where

$\psi_{\phi}(x_{1},\ldots,x_{n})=\neg\exists y\Big{(}\bigwedge_{i\in
E}\phi(x_{i},y)\wedge\bigwedge_{i\in\\{1,\ldots,n\\}\setminus
E}\neg\phi(x_{i},y)\Big{)}.$

(ii) We say that $M$ is eventually $NIP$ if every formula is eventually $NIP$
in $M$.

Note that (i) states that we have a special pattern that never exists. This
intuition helps to better understand the notion and how to use it further. In
the following, we will explain it better:

######  Remark 3.2.

(i) The subsequence $(a_{i})$ in the above is convergent for $\phi$.151515An
analysis of $\psi_{\phi}$ shows that the alternation number $n_{(a_{i})}$ of
$(\phi(a_{i},y):i<\omega)$ is finite. (See also Remark 3.5(ii) below.) Notice
that $n_{(a_{i})}$ depends on both the formula and the sequence; not just on
formula. This is a ‘wider’ notion of alternation number. Cf. [Kha19b], the
paragraph before Remark 2.11. That is, $(\phi(a_{i},b):i<\omega)$ converges
for any $b\in\cal U$. Moreover, $\psi_{\phi}(\bar{x})$ is in the Ehrenfeucht-
Mostovski type $EM((a_{i}))$ of $(a_{i})$.
(ii) In some sense, the notion of eventual $NIP$ is not new. In fact, a theory
$T$ is $NIP$ (in the usual sense) iff the monster model of $T$ is eventually
$NIP$ iff every model of $T$ is eventually $NIP$ iff some model $M$ of $T$ in
which all types over the empty set in countably many variables are realized is
eventually $NIP$. Cf. Proposition 2.14 in [Kha19b].
(iii) The notion of ‘eventual $NIP$’ is strictly stronger than the notion of
‘$NIP$ in a model’ in [KP18], and strictly weaker than the notion of ‘uniform
$NIP$ in a model’ in [Kha21].161616The directions are consequences of
definitions. Although, we strongly believe that the strictness holds, but we
have not found clear examples yet. Cf. Remark 3.7 and item (1) in the
“Concluding remarks/questions” below.

Let $X$ be a topological space and $f:X\to[0,1]$ be a function. Recall from
[Kha19b] that $f$ is called a difference of bounded semi-continuous functions
(short $DBSC$) if there exist bounded semi-continuous functions $F_{1}$ and
$F_{2}$ on $X$ with $f=F_{1}-F_{2}$. It is a well-known fact that, in general,
the class of $DBSC$ functions is a proper subclass of all Baire 1 function.
(Cf. [Kha19b], Section 2.)

We let $\phi^{*}(y,x)=\phi(x,y)$. Let $q=tp_{\phi^{*}}(b/M)$ be the function
$\phi^{*}(q,x):M\to\\{0,1\\}$ defined by $a\mapsto\phi^{*}(b,a)$. This
function is called a complete $\phi^{*}$-types over $M$. The set of all
complete $\phi^{*}$-types over $M$ is denoted by $S_{\phi^{*}}(M)$. We equip
$S_{\phi^{*}}(M)$ with the least topology in which all functions
$q\mapsto\phi^{*}(q,a)$ (for $a\in M$) are continuous. It is compact and
Hausdorff, and is totally disconnected.

######  Definition 3.3.

Let $p(x)$ be a global type which is finitely satisfiable in $M$.

(1) Suppose that $\phi(x,y)$ is a formula, and $p_{\phi}$ is the restriction
of $p$ to $\phi$-formulas. Define $f_{p}^{\phi}:S_{\phi^{*}}(M)\to\\{0,1\\}$
by $f_{p}^{\phi}(q)=1$ iff $\phi(x,b)\in p$ for some/any $b\models q$. We say
that:

(i) $p_{\phi}$ is definable over $M$ if $f_{p}^{\phi}$ is continuous,

(ii) $p_{\phi}$ is $DBSC$ definable over $M$ if $f_{p}^{\phi}$ is $DBSC$,

(iii) $p_{\phi}$ is Baire 1 definable over $M$ if $f_{p}^{\phi}$ is Baire 1.

(2) $p(x)$ is called ($DBSC$ or Baire 1) definable over $M$ iff for any
formula $\phi(x,y)$ the type $p_{\phi}$ is ($DBSC$ or Baire 1) definable over
$M$, respectively.

Notice that (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii) but in general (i)
$\nLeftarrow$ (ii) $\nLeftarrow$ (iii). (Cf. [Kha19b].)

######  Definition 3.4.

Let $(a_{i})\in\cal U$ be a sequence. We say that $(a_{i})$ is
$DBSC$-convergent (or $DBSC$-converges) if for any formula $\phi(x,y)$ there
is a natural number $N=N_{\phi}$ such that for any $b\in\cal U$,

$\sum_{i=1}^{\infty}|\phi(a_{i},b)-\phi(a_{i+1},b)|\leq N.$

In the following we explain the above notions and their relationship.

######  Remark 3.5.

(i) Notice that this notion is equivalent to having finite alternation number.
Although, this number depends on both the formula and the sequence; not just
on formulas.
(ii) Assuming that $(a_{i})$ is eventually indiscernible over the empty set
$\emptyset$. A sequence $(a_{i})$ is $DBSC$-convergent iff for any formula
$\phi(x,y)$ (without parameters) there is a formula
$\psi_{\phi}(x_{1},\ldots,x_{n})$, as be in Definition 3.1, such that
$\models\psi_{\phi}(a_{i_{1}},\ldots,a_{i_{n}})$ for any
$i_{1}<\cdots<i_{n}<\omega$ iff for any formula $\phi(x,y)$ (without
parameters) there is a natural number $n$ such that $\theta_{n,\phi}\notin
SEEM((a_{i})/\emptyset)$.171717$\theta_{n,\phi}$ was defined in the proof of
Theorem 2.11.
(iii) Suppose that $(a_{i})$ is $DBSC$-convergent. Then the sequence
$tp(a_{i}/{\cal U})$ converges. Moreover, for any formula $\phi(x,y)$ the
sequence $(tp_{\phi}(a_{i}/{\cal U}):i<\omega)$ converges to a type $p_{\phi}$
which is $DBSC$ definable (over any model $M\supseteq(a_{i})$).
(iv) Whenever $M$ is countable, $DBSC$-definability and strong Borel
definability (in the sense of [HP11]) are the same.

###### Proof.

(i) was first observed in [Kha19b]. (Cf. the paragraph before Remark 2.11 in
there.)
(ii) and (iii) follows form Lemma 2.8 of [Kha19b]. For the last part of (ii),
note that $\models\psi_{\phi}(a_{i_{1}},\ldots,a_{i_{n}})$ for any
$i_{1}<\cdots<i_{n}<\omega$, clearly implies that $\theta_{n,\phi}\notin
SEEM((a_{i})/\emptyset)$. For the converse, suppose that there are natural
numbers $n,N$ such that for any $N<i_{1}<\cdots<i_{n}$, $\theta_{n,\phi}\notin
SEEM((a_{i})/\emptyset)$. Then, we replace $(a_{i})$ by
$(a_{N+1},a_{N+2},\ldots)$ and use Ramsey’s theorem, if necessary.
(iv) was first mentioned in [Kha19b, Remark 2.15] and studied in [Kha21a]. ∎

We are ready to give a characterization of convergent Morley sequences over
countable models in the terms of eventual $NIP$.

######  Theorem 3.6.

Let $T$ be a countable theory and $M$ a countable model of $T$. Then the
following are equivalent:
(i) $M$ is eventually $NIP$.
(ii) For any $p(x)\in S(\cal U)$ which is finitely satisfiable in $M$, there
is a sequence $(c_{i}:i<\omega)\in M$ such that the sequence $(tp(c_{i}/{\cal
U}):i<\omega)$ $DBSC$-converges to $p$.
(iii) For any $p(x)\in S(\cal U)$ which is finitely satisfiable in $M$, there
is a Morley sequence $(d_{i}:i<\omega)$ of $p$ over $M$ such that
$(tp(d_{i}/{\cal U}):i<\omega)$ converges.

###### Proof.

(iii) $\Rightarrow$ (i): Let $\phi(x,y)$ be a formula, and $(c_{i})$ a
sequence in $M$. Let $p$ be an accumulation point of $\\{tp(c_{i}/{\cal
U}):i\in\omega\\}$. (Therefore, $p$ is finitely satisfiable in $M$.) Let
$I=(d_{i})$ be a Morley sequence of $p$ over $M$. By (iii), $(tp(d_{i}/{\cal
U}):i<\omega)$ converges.

Claim: There is a subsequence $(a_{i})$ of $(c_{i})$ such that $\lim
tp(a_{i}/MI)=p|_{MI}$.181818Compare the argument of Lemma 2.7(ii) above. An
alternative (and more model-theoretic) argument can be given that is similar
to the one Gannon gave in the first paragraph of the proof of [G21, Lemma
4.6].

_Proof_ : The closure of $\\{tp(c_{i}/MI):i<\omega\\}\subset S_{x}(MI)$ is
second-countable and compact, and so metrizable. Therefore, there is a
sequence $(a_{i})\in\\{c_{i}:i<\omega\\}$ such that $\lim
tp(a_{i}/MI)=p|_{MI}$. We can assume that $(a_{i})$ is a subsequence of
$(c_{i})$. (If not, consider a subsequence of $(a_{i})$ which is a subsequence
of $(c_{i})$.) $\dashv_{\text{claim}}$

By Fact 2.6, we can assume that $(a_{i})$ is eventually indiscernible. Now, by
Lemma 2.7, $SEEM((a_{i})/M)=SEM((d_{i})/M)$. By Remark 3.5, as $(d_{i})$
converges, this means that the condition (i) of Definition 3.1 holds for
$(c_{i})$ and $\phi(x,y)$. (Cf. Remark 2.12(iv).)
(ii) $\Rightarrow$ (iii): Let $p(x)\in S(\cal U)$ be finitely satisfied in
$M$, and $(c_{i}:i<\omega)\in M$ such that the sequence $(tp(c_{i}/{\cal
U}):i<\omega)$ $DBSC$-converges to $p$. Recall that some Morley sequence of
$p$ over $M$ is convergent if and only if every Morley sequence of $p$ over
$M$ is convergent. Let $I=(d_{i})$ be a Morley sequence in $p$ over $M$. By
(ii), $\lim tp(c_{i}/MI)=p|_{MI}$, and so by Lemma 2.7, we have
$SEEM((c_{i})/M)=SEM((d_{i})/M)$. As $(tp(c_{i}/{\cal U}):i<\omega)$ is
$DBSC$-convergent, by Remark 3.5, $(d_{i})$ converges.
(i) $\Rightarrow$ (ii): Let $p(x)\in S(\cal U)$ be finitely satisfied in $M$.
By Lemma 2.7, there are a sequence $(c_{i})$ in $M$ and a Morley sequence
$(d_{i})$ of $p$ over $M$ such that $tp(c_{i}/M\cup(d_{i}))\to
p|_{M\cup(d_{i})}$ and $SEEM((c_{i})/M)=SEM((d_{i})/M)$. By (i), as $T$ is
countable, using a diagonal argument, there is a subsequence
$(c_{i}^{\prime})\subseteq(c_{i})$ such that $(tp(c_{i}^{\prime}/{\cal
U}):i<\omega)$ $DBSC$-converges. Therefore, using an argument similar to the
proof of Claim 2 in Theorem 2.11 (or directly), we can see that the Morley
sequence $(d_{i})$ is convergent. By an argument similar to Theorem 2.11,
$(tp(c_{i}^{\prime}/{\cal U}):i<\omega)$ converges to $p$. (Cf. Remark
2.12(iv).) ∎

######  Remark 3.7.

Let $T$ be a countable theory and $M$ a countable model of it. Suppose that
any $p(x)\in S(\cal U)$ which is finitely satisfied in $M$ is $DBSC$ definable
over $M$. In this case, using the BFT theorem, it is easy to show that for any
such type $p(x)$ there is a sequence $(c_{i})\in M$ such that $\lim
tp(c_{i}/{\cal U})=p$. Notice that there is no reason that $(tp(c_{i}/{\cal
U}):i<\omega)$ $DBSC$-converges to $p$. A question arises. With the above
assumptions, for any $p(x)\in S(\cal U)$ which is finitely satisfied in $M$,
is there any sequence $(c_{i})\in M$ such that $(tp(c_{i}/{\cal U}):i<\omega)$
$DBSC$-converges to $p$? We believe that the answer is negative, although we
have not found a counterexample yet.

### An application to definable groups

To finish this section, we give an example where the notion of eventual $NIP$
is used to deduce results about definable groups.

######  Lemma 3.8.

Let $G$ be a definable group. Let $p,q$ be invariant types concentrating on
$G$ such that both $p_{x}\otimes q_{y}$ and $q_{y}\otimes p_{x}$ imply $x\cdot
y=y\cdot x$.191919Recall that a type $p(x)$ concentrates on a group $G$ if
$p\vdash x\in G$. If some/any Morley sequence of $p$ converges, then $a\cdot
b=b\cdot a$ for any $a\models p$ and $b\models q$.

###### Proof.

The proof is an adaptation of [S15, Lemma 2.26]. By compactness, there is a
small model $M$ such that $p,q$ are $M$-invariant and for any $(a,b)$
realizing one of $(p\otimes q)|_{M}$ or $(q\otimes p)|_{M}$ we have $a\cdot
b=b\cdot a$.

We claim that there is no infinite sequence $(a_{n}b_{n}:n<\omega)$ such that
$a_{n}\models p|_{Ma_{<n}b_{<n}}$, $b_{n}\models q|_{Ma_{<n}b_{<n}}$ and
$a_{n}\cdot b_{n}\neq b_{n}\cdot a_{n}$. If not, by hypothesis $a_{n}\cdot
b_{m}=b_{m}\cdot a_{n}$ for $n\neq m$. For any $I\subset\omega$ finite, define
$b_{I}=\prod_{n\in I}b_{n}$. Therefore, $a_{n}\cdot b_{I}=b_{I}\cdot a_{n}$ if
and only if $n\notin I$. This means that the sequence
$(\phi(a_{n},y):N<\omega)$ does not converges where $\phi(a_{n},y):=a_{n}\cdot
y\neq y\cdot a_{n}$. As $(a_{n}:n<\omega)$ is a Morley sequence of $p$, this
contradicts the assumption.

Therefore, by the above claim, there is some $n$ such that any sequence with
the above construction has the length smaller than $n$. Let
$p_{0}=p|_{Ma_{<n}b_{<n}}$ and $q_{0}=q|_{Ma_{<n}b_{<n}}$. Then
$p_{0}(x)\wedge q_{0}(y)\rightarrow x\cdot y=y\cdot x$. ∎

######  Proposition 3.9.

Let $T$ be a countable theory and $G$ a definable group. Assume that there is
a countable subset $A\subset G$ such that any two elements of $A$ commute, and
$A$ is eventually $NIP$. Then there is a definable abelian subgroup of $G$
containing $A$.

###### Proof.

Let $S_{A}\subset S({\cal U})$ be the set of global $1$-types finitely
satisfible in $A$. Notice that, as $A$ is eventually $NIP$, by Theorem 3.6,
the Morley sequence of any type in $S_{A}$ is convergent. Therefore, for any
$p,q\in S_{A}$, the pair $(p,q)$ satisfies the hypothesis of Lemma 3.8 above.
The rest is similar to the argument of Proposition 2.27 of [S15]. Indeed, by
Lemma 3.8 and compactness, one can find formulas $\phi(x)$ and $\psi(y)$ such
that $\phi(x)\wedge\psi(y)\rightarrow x\cdot y=y\cdot x$ and all types of
$S_{A}$ concentrate on both $\phi(x)$ and $\psi(y)$. Set
$H:=C_{G}(C_{G}(\phi\wedge\psi))$, where $C_{G}(X)=\\{g\in G:g\cdot x=x\cdot
g\text{ for all }x\in X\\}$. Then $H$ is a definable abelian subgroup of $G$
containing $A$. ∎

######  Remark 3.10.

(1) Notice that, if any two elements of a set $A$ commute, then
$C_{G}(C_{G}(A))$ is abelian, but $C_{G}(A)$ is not automatically abelian
(even when $A$ is a subgroup).202020Let $G$ be any non-abelian group, and let
$e$ be the identity of the group. Then $C_{G}(e)=G$ is non-abelian. In the
following, we provide a proof:
Notice that, as any two elements of $A$ commute, $A\subseteq C_{G}(A)$.
Therefore $C_{G}(A)\supseteq C_{G}(C_{G}(A))$.212121Recall that for any
$X\subseteq Y$, $C_{G}(Y)\supseteq C_{G}(X)$. Let $a,b\in C_{G}(C_{G}(A))$.
Since $b\in C_{G}(A)$, so by definition $ab=ba$. As $a,b$ are arbitrary,
$C_{G}(C_{G}(X))$ is abelian.222222This short statement was suggested to us by
Narges Hosseinzadeh.
(2) In Proposition 3.9, if $A$ is finite, we don’t need eventual $NIP$: take
$H=C_{G}(C_{G}(A))$.

##  4 Generically stable types

Here we want to give new characterizations of generically stable types for
countable theories. The notion of generically stable types in general theories
was introduced in [PT11]. Recall from [CG20, Prop 3.2] that a global type $p$
is generically stable over a small set $A$ if $p$ is $A$-invariant and for any
Morley sequence $(a_{i}:i<\omega)$ of $p$ over $A$, we have $\lim
tp(a_{i}/{\cal U})=p$.

Before giving the results let us recall that:

######  Fact 4.1 ([G21], Fact 2.6).

Let $M$ be small set, and $p(x)\in S({\cal U})$ a global $M$-invariant type.
(i) If $p$ is generically stable over $M$, then $p$ is definable over and
finitely satisfiable in $M$.
(ii) If $p$ is generically stable over $M$ and $M_{0}$-invariant, then $p$ is
generically stable over $M_{0}$. If $p$ is definable over and finitely
satisfiable in $M$ and $M_{0}$-invariant, the same holds.
(iii) Assuming that $T$ is countable, if $p$ is generically stable over $M$,
there exists a countable elementary substructure $M_{0}$ such that $p$ is
generically stable over $M_{0}$. The same holds for definable and finitely
satisfiable case.

######  Lemma 4.2.

Let $T$ be a (countable or uncountable) theory, $A\subset{\cal U}$, and
$p(x)\in S({\cal U})$ a global $A$-invariant type. Suppose that some/any
Morley sequence of $p$ is totally indiscernible, AND some/any Morley sequence
of $p$ is convergent. Then $p$ is generically stable.232323This was first
announced in Remark 3.3(iii) of [HP11].

###### Proof.

Let $I=(a_{i})$ be a Morley sequence of $p$ over $A$. We show that $\lim
tp(a_{i}/{\cal U})=p$. Let $\phi(x,b)\in p$ and $J\models
p^{(\omega)}|_{AIb}$. Set $I_{n}=(a_{1},\ldots,a_{n})$ for all $n$. Notice
that all points of $J$ satisfy $\phi(x,b)$, and $I_{n}+J$ is a Morley sequence
(for all $n$).242424$I_{n}+J$ is the concatenation of $I_{n}$ and $J$. It has
$I_{n}$ as initial segment and $J$ as the complementary final segment. We
claim that at most a finite number of points of $I$ satisfy $\neg\phi(x,b)$.
If not, for each $k$, there is a natural number $n_{k}$ such that
$\\#\\{a_{i}\in I_{n_{k}}:\models\neg\phi(a_{i},b)\\}\geq k$. As $I_{n}+J$ is
totally indiscernible (for all $n$), this implies that for each $n$,
$\theta_{n,\phi}(x_{1},\ldots,x_{n})\in tp(J)$ where

$\theta_{n,\phi}(x_{1},\ldots,x_{n})=\forall F\subseteq\\{1,\ldots,n\\}\exists
y_{F}\Big{(}\bigwedge_{i\in F}\phi(x_{i},y_{F})\wedge\bigwedge_{i\notin
F}\neg\phi(x_{i},y_{F})\Big{)}.$

(Recall that $\theta_{n,\phi}$ was introduced in the proof of Theorem 2.11.
Notice that if $\\#\\{i:\models\phi(a_{i},b)\\}=\aleph_{0}$ then we do not
need total indiscernibility, but only indiscernibility.) Equivalently, $J$ is
not convergent, a contradiction. ∎

######  Remark 4.3.

Let $T$ be a (countable or uncountable) theory, $A\subset{\cal U}$, and
$p(x)\in S({\cal U})$ a global $A$-invariant type. The following are
equivalent.
(i) $p$ is generically stable.
(ii) $p$ is definable over a small model AND there is a Morley sequence
$(a_{i}:i<\omega)$ of $p$ over $A$ such that $\lim tp(a_{i}/{\cal U})=p$.

###### Proof.

(i) $\Longrightarrow$ (ii) follows from Fact 4.1. (Cf. [S15], Theorem 2.29.)

(ii) $\Longrightarrow$ (i): Suppose that there is a Morley sequence
$I=(a_{i})$ of $p$ over $A$ such that $\lim I=p$. As $p$ is definable and
finitely satisfiable, some/any Morley sequence of $p$ is totally
indiscernible. (Cf. [Kha21, Corollary 4.11] for a proof that any definable and
finitely satisfiable type commutes with itself and a generalization to
measures.) Therefore, by Lemma 4.2, $p$ is generically stable. ∎

The following theorem gives new characterizations of generically stable types
for countable theories. The important ones to note immediately are (ii) and
(v).

######  Theorem 4.4.

Let $T$ be a countable theory, $M$ a small model of $T$, and $p(x)\in S({\cal
U})$ a global $M$-invariant type. The following are equivalent:
(i) $p$ is generically stable over $M$.
(ii) $p$ is definable over a small model, AND there is a sequence $(c_{i})$ in
$M$ such that $(tp(c_{i}/{\cal U}):i<\omega)$ $DBSC$-converges to $p$.
(iii) $p$ is definable over and finitely satisfiable in some small model, AND
there is a convergent Morley sequence of $p$ over $M$.
(iv) $p$ is definable over a small model, AND there is a Morley sequence
$(a_{i})$ of $p$ over $M$ such that $\lim tp(a_{i}/{\cal U})=p$.
Suppose moreover that $T$ has $NSOP$, then each of (v), (vi), (vii) below is
also equivalent to (i), (ii), (iii), (iv) above:
(v) There is a sequence $(c_{i})$ in $M$ such that $(tp(c_{i}/{\cal
U}):i<\omega)$ $DBSC$-converges to $p$.
(vi) $p$ is finitely satisfiable in a countable model $M_{0}\prec M$, AND
there is a convergent Morley sequence of $p$ over $M$.
(vii) There is a Morley sequence $(a_{i})$ of $p$ over $M$ such that $\lim
tp(a_{i}/{\cal U})=p$.

###### Proof.

(i) $\Longrightarrow$ (ii): As $T$ is countable, by Fact 4.1, we can assume
that $p$ is generically stable over a countable substructure $M_{0}\prec M$.
By Corollary 2.13, there is a sequence $(c_{i})$ in $M$ such that
$(tp(c_{i}/{\cal U}):i<\omega)$ converges to $p$. Notice that in the proof of
Theorem 2.11 for any formula $\phi$ there is a natural number $n$ such that
the formula $\theta_{n,\phi}$ does not belong to $SEEM((c_{i})/M)$. This means
that $(tp(c_{i}/{\cal U}):i<\omega)$ is $DBSC$-convergent.

(ii) $\Longrightarrow$ (i): Clearly, $p$ is finitely satisfiable in $M$. As
$p$ is definable and finitely satisfiable, any Morley sequence of $p$ is
totally indiscernible. (Cf. [Kha21, Corollary 4.11].) Let $(d_{i})$ be a
Morley sequence of $p$ over $M$. By Fact 2.6, we can assume that $(c_{i})$ is
eventually indiscernible over $M\cup(d_{i})$. By Lemma 2.7, it is easy to see
that $SEEM((c_{i})/M)=SEM((d_{i})/M)$. Therefore, as $(c_{i})$ is
$DBSC$-convergent, the Morley sequence $(d_{i})$ converges. By Lemma 4.2, $p$
is generically stable.

(iii) $\Longrightarrow$ (i) follows from Lemma 4.2 and the fact that the
Morley sequences of definable and finitely satisfiable types are totally
indiscernible.

(i) $\Longrightarrow$ (iii) follows from the direction (i) $\Longrightarrow$
(ii) of [CG20, Pro. 3.2]. (Recall that generically stable types are definable
and finitely satisfiable.)

(iv) $\iff$ (i) follows from Remark 4.3.

The directions (ii) $\Longrightarrow$ (v) and (iii) $\Longrightarrow$ (vi) and
(iv) $\Longrightarrow$ (vii) are evident (and hold in any theory).

For the rest of the proof, suppose moreover that $T$ has $NSOP$.

Then, (v) $\Longrightarrow$ (ii) follows from Proposition 2.10 of [Kha19b] and
the Eberlein–Grothendieck criterion ([Kha19b, Fact 2.2]). Indeed, by the
direction (i) $\Longrightarrow$ (iv) of [Kha19b, Pro. 2.10], for any formula
$\phi(x,y)$, there is no infinite sequence $(b_{j})$ such that
$\phi(c_{i},b_{j})$ holds iff $i<j$. By Fact 2.2 of [Kha19b], this means that
the limit of $(\phi(c_{i},y):i<\omega)$ is a continuous function.
Equivalently, $p$ is definable over $M$. (See also Remark 2.11 of [Kha19b].)

(vi) $\Longrightarrow$ (iii): Suppose that $p$ is finitely satisfiable in
$M_{0}\prec M$ with $|M_{0}|=\aleph_{0}$. By Theorem 2.11, there is a sequence
$(c_{i})\in M_{0}$ such that $(tp(c_{i}/{\cal U}):i<\omega)$ $DBSC$-converges
to $p$. By the direction (i) $\Longrightarrow$ (iv) of [Kha19b, Pro. 2.10] and
[Kha19b, Fact 2.2], $p$ is definable over $M_{0}$. Therefore, (iii) holds.

(vii) $\Longrightarrow$ (iv): As $(a_{i})$ is indiscernible and convergent,
the sequence $(tp(a_{i}/{\cal U}):i<\omega)$ is $DBSC$-convergent. This means,
by $NSOP$ (i.e. the direction (i) $\Longrightarrow$ (iv) of [Kha19b, Pro.
2.10] and [Kha19b, Fact 2.2]), that $p$ is definable. ∎

######  Remark 4.5.

(i) It is not hard to give a variant of Theorem 4.4 for uncountable theories.
Indeed, we can consider all countable fragments of the languages, and use the
above argument.
(ii) With the assumption of Theorem 4.4, then $(*)$ below is also equivalent
to (i)—(iv) in Theorem 4.4:

$(*)$ For any $B\supset M$, $p$ is the unique global nonforking extension of
$p|_{B}$, AND there is a convergent Morley sequence of $p$ over $M$.

The argument is an adaptation of the proof of [HP11, Proposition 3.2]. See
also Proposition 4.6(ii) below.

As the referee pointed out to us, the following proposition is not
new.252525(i) is Remark 5.18 of [CGH21], and (ii) follows from the fact that
generically stable types are stationary (cf. [PT11, Proposition 1(iv)]).
Although for the sake of completeness we give a proof using the above
observations.

######  Proposition 4.6.

Let $T$ be a (countable or uncountable) theory and $p$ a generically stable
type.
(i) For any invariant type $q$, $p\otimes q=q\otimes p$.
(ii) If $p$ is $A$-invariant, then $p$ is the unique $A$-invariant extension
of $p|_{A}$.

###### Proof.

(i) follows from the argument of Proposition 2.33 of [S15] by replacing [S15,
Lemma 2.28] with the argument of Lemma 4.2 above. Indeed, suppose for a
contradiction that for some formula $\phi(x,y,c)\in L({\cal U})$ (where $c$ is
a tuple of elements) we have $p_{x}\otimes q_{y}\vdash\phi(x,y,c)$ and
$q_{y}\otimes p_{x}\vdash\neg\phi(x,y,c)$. Let $(a_{i}:i<\omega)\models
p^{(\omega)}$, $b\models q|_{{\cal U}a_{<\omega}}$ and $(a_{i}:\omega\leq
i<\omega 2)\models p^{(\omega)}|_{{\cal U}a_{<\omega}b}$. Then for $i<\omega$,
$\neg\phi(a_{i},b,c)$ holds and for $i\geq\omega$, we have $\phi(a_{i},b,c)$.
(Recall the definition of Morley products in 2.2.1 of [S15].) As
$(a_{i}:i<\omega 2)$ is totally indiscernible, similar to the argument of
Lemma 4.2, it is easy to verify that for each $n$,
$\theta_{n,\phi}(x_{1},\ldots,x_{n})\in tp((a_{i})/\emptyset)$ where

$\theta_{n,\phi}(x_{1},\ldots,x_{n})=\forall F\subseteq\\{1,\ldots,n\\}\exists
y_{F}\exists y_{c}\Big{(}\bigwedge_{i\in
F}\phi(x_{i},y_{F},y_{c})\wedge\bigwedge_{i\notin
F}\neg\phi(x_{i},y_{F},y_{c})\Big{)}.$

Equivalently, the sequence $(\phi(a_{i},y_{F},y_{c}):i<\omega)$ is not
convergent, a contradiction.

(ii): Let $q$ be any $A$-invariant extension of $p|_{A}$.

Claim: $p^{(\omega)}|_{A}=q^{(\omega)}|_{A}$.

_Proof_ : The proof is by induction, and similar to the argument of
Proposition 2.35 of [S15]. The base case is $p|_{A}=q|_{A}$. The induction
hypothesis is that $p^{(n)}|_{A}=q^{(n)}|_{A}$. Using (i) above and
associativity of Morley products, we have:

$\displaystyle q^{(n+1)}_{x_{1},\ldots,x_{n+1}}|_{A}$
$\displaystyle=(q_{x_{n+1}}\otimes q^{(n)}_{x_{1},\ldots,x_{n}})|_{A}$
$\displaystyle=(q_{x_{n+1}}\otimes p^{(n)}_{x_{1},\ldots,x_{n}})|_{A}$
$\displaystyle\stackrel{{\scriptstyle(*)}}{{=}}(p^{(n)}_{x_{1},\ldots,x_{n}}\otimes
q_{x_{n+1}})|_{A}$ $\displaystyle=(p^{(n)}_{x_{1},\ldots,x_{n}}\otimes
p_{x_{n+1}})|_{A}$ $\displaystyle=p^{(n+1)}_{x_{1},\ldots,x_{n+1}}|_{A}.$

Notice that (i) and associativity of Morley products are used in
$(*)$.262626Notice that we can not use Lemma 2.34 of [S15], because it is not
known whether the products of generically stable types are generically stable
or not. Although, the associativity of Morley products and the part (i) of
Proposition 4.6 are sufficient here. $\dashv_{\text{claim}}$

Therefore, every Morley sequence of $q$ is totally indiscernible AND
convergent. By Lemma 4.2, $q$ is generically stable and so $\lim I=q$ for any
Morley sequence of $q$. This means that $p=\lim I=q$ for any
$I=p^{(\omega)}|_{A}=q^{(\omega)}|_{A}$. (Alternatively, as
$p^{(\omega)}|_{A}=q^{(\omega)}|_{A}$, one can use Remark 2.12(ii) above.) ∎

Here we want to give a local version of a classical result [HP11, Proposition
3.2]:

######  Theorem 4.7.

Let $T$ be a (countable or uncountable) theory, $M$ be a model of $T$, and
$p(x)$ a global $M$-invariant type. Suppose that there is an elementary
extension $M^{\prime}\succ M$ containing a Morley sequence of $p$ such that
$M^{\prime}$ is eventually $NIP$. Then the following are equivalent.

(i) $p=\lim tp(a_{i}/{\cal U})$ for any $(a_{i})\models p^{(\omega)}|_{M}$.

(ii) $p$ is definable over and finitely satisfiable in $M$.

(iii) $p_{x}\otimes p_{y}=p_{y}\otimes p_{x}$.

(iv) any Morley sequence of $p$ is totally indiscernible.

###### Proof.

(i) $\Longrightarrow$ (ii) $\Longrightarrow$ (iii) $\Longrightarrow$ (iv) are
standard and hold in any theory. (Cf. Theorem 2.29 of [S15].)

(iv) $\Longrightarrow$ (i): Let $J\in M^{\prime}$ be a Morley sequence of $p$.
Since $M^{\prime}$ is eventually $NIP$, the sequence $J$ is convergent. By
Lemma 4.2, $p$ is generically stable. ∎

Notice that the above theorem holds with a weaker assumption, namely every
formula has $NIP$ in $M^{\prime}$. (Cf. [KP18], for definition of $NIP$ in a
model.) This easily follows from indiscernibility of Morley sequences.

### Eventually stable models

The story started from Grothendieck’s double limit characterization of weak
relative compactness, Theorem 6 in [G52]. In [Ben14] Ben Yaacov showed that
the “Fundamental Theorem of Stability” is in fact a consequence of
Grothendieck’s theorem. Shortly afterwards, Pillay [P18] pointed out that the
model-theoretic meaning of the Grothendieck theorem is that the formula
$\phi(x,y)$ does not have the order property in $M$ if and only if every
complete $\phi$-type $p(x)\in S_{\phi}(M)$ has an extension to a complete type
$p^{\prime}\in S_{\phi}({\cal U})$ which is finitely satisfiable in, and
definable over $M$. There, he called such types ‘generically stable’ and said:
“We will investigate later to what extent we can deduce the stronger notions
of generic stability from not the order property in $M$”. Here, using the
previous results/observations, we can prove a result similar to [P18] for the
stronger notions of generic stability. Maybe the following result is the end
of this story, and of course the beginning of another story.

######  Definition 4.8.

Let $M$ be a model. (i) We say that $M$ has no order if for any formula
$\phi(x,y)$ there do not exist $(a_{i}),(b_{i})$ in $M$ for $i<\omega$ such
that $M\models\phi(a_{i},b_{j})$ iff $i\leq j$.
(ii) We say that $M$ is eventually stable if

(1) $M$ has no order, and

(2) $M$ is eventually $NIP$ (as in Definition 3.1).

######  Remark 4.9.

(i) In stable theories, every model is eventually stable.
(ii) In $NIP$ theories, every model which has no order is eventually stable.

###### Proof.

(i): If not, similar to the argument of (i) $\Rightarrow$ (iii) of [Kha19b,
Proposition 2.14], we can find a formula $\phi(x,y)$, an indiscernible
sequence $(c_{i})$, and an element $d$ such that $\phi(c_{i},d)$ holds if and
only if $i$ is even. This contradicts $NIP$.

(ii) Suppose that the theory $T$ is $NIP$ and $M\models T$ has no order.
Suppose for a contradiction that $M$ is not eventually $NIP$. Similar to (i),
we can find a formula $\phi(x,y)$, an indiscernible sequence $(c_{i})$
(possibly in an elementary extension of $M$), and an element $d\in\cal U$ such
that $\phi(c_{i},d)$ holds if and only if $i$ is even, a contradiction. ∎

######  Theorem 4.10.

Let $T$ be a (countable or uncountable) theory, and $M$ be a model of $T$. The
following are equivalent:
(i) $M$ is eventually stable.
(ii) Any type $p\in S_{x}(M)$ has an extension to a global type $p^{\prime}\in
S_{x}({\cal U})$ which is generically stable over $M$.

###### Proof.

First, without loss of generality we can assume that $T$ is countable.272727We
consider all countable fragments of the languages. By Proposition 2.3(c) of
[P18], $M$ has no order if and only if any type $p\in S_{x}(M)$ has an
extension to a global type $p^{\prime}\in S_{x}({\cal U})$ which is finitely
satisfiable in, and definable over $M$.282828In fact, we do not need to use
Grothendieck’s argument. Indeed, assuming eventual $NIP$, as any Morley
sequence is controlled by a sequence in the model and vice versa (cf. Theorem
3.6), we can use the standard fact that a Morley sequence is totally
indiscernible iff it has no order (cf. Theorem 12.37 of [Poi00]). By Theorems
3.6 and 4.4 above, any global type $p^{\prime}\in S_{x}({\cal U})$ which is
finitely satisfiable in, and definable over $M$ is generically stable over $M$
if and only if $M$ is eventually $NIP$. This proves the theorem. ∎

### Concluding remarks/questions

(1) In Example 2.18 of [Kha19b], we built a graph $N$ with the following
property: (i) there is a sequence $(a_{i})\in N$ such that $R(a_{i},y)$
converges, and (ii) $R(a_{i},y)$ is not $DBSC$-convergent. We guess that a
modification of this example leads to a definable type $p$ such that: (i)
there is a sequence $(a_{i})$ with $\lim tp(a_{i}/{\cal U})=p$, and (ii) $p$
is not the limit of any $DBSC$-convergent sequence. (For this, one need to
remove the axiom schema (1) in Example 2.18, and to check the above
properties.) Therefore, by Theorem 4.4, $p$ is not generically stable. This
approach probably answer to Question 4.15 of [G21].

(2) These results/observations can be generalize to “continuous logic”
[BBHU08]. On the other hand, one can generalize Theorem 3.6 for measures in
classical logic. This is a generalization of another result of Gannon [G21,
Theorem 5.10]. Recall that measures in classical logic correspond to types in
continuous logic. This means that a generalization of Theorem 3.6 to
continuous logic leads to a generalization of this theorem for measures in
classical logic, and vice versa.

(3) In [Kha21], we claimed that in the language of Banach spaces in continuous
logic, there is a Krivine-Maurey type theorem for $NIP$ theories (or even
$NIP$ spaces). That is, for any separable $NIP$ space $X$ there exists a
spreading model of $X$ containing $c_{0}$ or $\ell_{p}$ for some $1\leq
p<\infty$. We believe that the results/observations of the present paper are
sufficient tools and they lead to a proof of this conjuncture. For example,
notice that $EEM$-types correspond to spreading models in Banach space theory.
On the other hand, types of $c_{0}$ and $\ell_{p}$ are symmetric in a strong
sense. Finally, the types of $c_{0}$ or $\ell_{p}$ are finitely satisfied in
any Banach space, by Krivine’s theorem.

(4) In [Kha21], we showed that every $\aleph_{0}$-categotical Banach space
contains $c_{0}$ or $\ell_{p}$. What is the translation of this observation
into “classical logic” (if such a translation is essentially possible)?
Similar questions can be asked about the Krivine-Maurey theorem (and the claim
in (3) whenever a proof of it is given).

We will study them elsewhere. (See for example [Kha22], for (2).)

Acknowledgements. I want to thank Predrag Tanović for reading a version of
this article and for his helpful comments (especially because of the argument
of Remark 2.12(iv)).

I would like to thank the Institute for Basic Sciences (IPM), Tehran, Iran.
Research partially supported by IPM grant 1400030118.

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