# Simple Homogeneous Structures and Indiscernible Sequence Invariants

John T. Baldwin  ,  James Freitag  and  Scott Mutchnik  Department of
Mathematics, Statistics, and Computer Science, University of Illinois at
Chicago<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We introduce some properties describing dependence in indiscernible sequences:
$F_{ind}$ and its dual $F_{Mb}$, the definable Morley property, and
$n$-resolvability. Applying these properties, we establish the following
results:

We show that the degree of nonminimality introduced by Freitag and Moosa,
which is closely related to $F_{ind}$ (equal in $\mathrm{DCF}_{0}$), may take
on any positive integer value in an $\omega$-stable theory, answering a
question of Freitag, Jaoui, and Moosa.

Proving a conjecture of Koponen, we show that every simple theory with
quantifier elimination in a finite relational language has finite rank and is
one-based. Our proofs of the supersimple case use Tomasić and Wagner’s results
on pseudolinearity, and the forking degrees $F_{\lambda}$ of Freitag. The
arguments closely rely on producing a type $q$ with $F_{Mb}(q)=\infty$. Our
proof of the strictly simple case uses our notion of $n$-resolvability (itself
related to $F_{Mb}$) which generalizes the $\mathrm{SU}_{(p)}$-rank of a
canonical base to the non-supersimple case. In this setting, both $F_{Mb}$ and
$n$-resolvability allow us to distinguish between infinitely many
indiscernible sequences over a finite set, contradicting quantifier
elimination in a finite relational language.

We prove some variants of the simple Kim-forking conjecture, a generalization
of the stable forking conjecture to $\mathrm{NSOP}_{1}$ theories. We show a
global analogue of the simple Kim-forking conjecture with infinitely many
variables holds in every $\mathrm{NSOP}_{1}$ theory, and show that Kim-forking
with a realization of a type $p$ with $\mathrm{F}_{Mb}(p)<\infty$ satisfies a
finite-variable version of this result. We then show, in a low
$\mathrm{NSOP}_{1}$ theory or when $p$ is isolated, if $p\in S(C)$ has the
definable Morley property for Kim-independence, Kim-forking with realizations
of $p$ gives a nontrivial instance of the simple Kim-forking conjecture
itself. In particular, when $F_{Mb}(p)<\infty$ and
$|S^{F_{Mb}(p)+1}(C)|<\infty$, Kim-forking with realizations of $p$ gives us a
nontrivial instance of the simple Kim-forking conjecture.

We show that the quantity $F_{Mb}$, motivated in simple and
$\mathrm{NSOP}_{1}$ theories by the above results, is in fact nontrivial even
in stable theories. Using constructions based on the free projective planes
studied by Hyttinen and Paolini, for every $n<\infty$, we find a stable theory
admitting types $p$ such that $F_{Mb}(p)=n$, and so that $F_{Mb}(p)\leq n$ for
all types $p$.

JB was partially supported by Simons travel grant G3535. JF was partially
supported by NSF CAREER award number 1945251. SM was supported by the NSF
under Grant No. DMS-2303034.

###### Contents

  1. 1 Introduction
    1. 1.1 Nonminimality
    2. 1.2 Countable categoricity and the Koponen conjecture
    3. 1.3 Stable forking and generalizations
    4. 1.4 Overall Organization of the paper
  2. 2 Degrees, nonminimality, and invariants of indiscernibility
    1. 2.1 Degree of nonminimality
    2. 2.2 Related notions involving canonical bases
  3. 3 Basic properties of the invariants
    1. 3.1 Basic Examples
    2. 3.2 Inequalities and basic results
  4. 4 $F_{Mb}$ can be finite, but larger than $1$, in a stable theory
  5. 5 $\operatorname{nmdeg}$ and $F_{ind}$ in categorical theories
    1. 5.1 Hrushovski constructions
    2. 5.2 Supersimple and strictly stable $\aleph_{0}$-categorical theories
    3. 5.3 $\aleph_{1}$-categorical theories
  6. 6 The Koponen conjecture
  7. 7 Definability of Morley sequences and the simple Kim-forking conjecture

##  1\. Introduction

Forking, dividing, and their associated notions of independence are at the
center of modern model theory from both a theoretical perspective and in
applications. Many of the most important dividing lines in model theory are
most easily defined by the presence or absence of certain local combinatorical
properties, but many admit equivalent characterizations by properties of
independence. This completely different perspective often plays a powerful
role in results and gives anyone who studies the subject a feeling that many
of the prominent dividing lines are canonical - they arise inevitably from a
number of different perspectives. For instance, Shelah [81] introduced _simple
theories_ as those in which a certain combinatorial configuration (the tree
property) was absent. The class is equivalently characterized by the property
that each type does not fork over a subset of its domain of size at most $|T|$
[38]. Kim and Pillay [52] gave another characterization, proving under a list
of general assumptions on an abstract independence relation
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{0}$ that the relation must be nonforking
independence in a simple theory. In a more applied direction, there is for
instance, a large literature whose central goal is characterizing forking
dependence for realizations of a specific type (or class of types). For
instance, there is extensive recent work just in the theory of differentially
closed fields [31, 36, 37, 69].

In a general theory, dividing of any partial type can always be detected by
some indiscernible sequence in the parameters of the formulas. _The central
thrust of this paper is to make quantitative various aspects of these
indiscernible sequences which control forking and dividing._ We do so by
defining several new invariants of indiscernible sequences relative to a
theory, all of which are closely related to how forking is witnessed in the
theory. We believe many of these invariants are of intrinsic interest - some
are already studied in the literature from a different perspective. For
instance, Lemma 3.2.1 is a generalization of the most natural case111In the
case that the outputs of system are the solutions themselves, the parameters
to be identified are the coefficients, and the coefficients form a canonical
base of the generic solution of the main result of [72, Theorem 3.1], a result
about the number of independent observations needed to determine the value of
the coefficients of a system given by differential equations with a fixed
model. After developing basic results around our invariants and connections
between them, we turn towards a number of applications, which we describe
next.

###  1.1. Nonminimality

The inspiration for our invariants comes from several sources. Several of our
notions are closely related to the _degree of nonminimality_ of a type [33].
Given a type $p$ of Lascar rank larger than $1$, $\operatorname{nmdeg}{(p)}$
is defined222In Section 2 there are careful definitions of the invariants we
study. to be the minimal number of realizations of the type $p$ such that $p$
has a nonalgebraic forking extension over the parameter set given by these
realizations.

Powerful upper bounds for the degree of nonminimality were established in
theories satisfying various hypotheses [33], which resulted in several
applications [32, 22]. But as the authors of [35] write:

> Concerning lower bounds on the degree of nonminimality, the situation is
> highly unsatisfactory. Examples of degree $>1$ seem very difficult to
> produce. In fact, we were unable to construct any of degree strictly greater
> than $2$.

We resolve this problem, first in the setting of simple theories in Section 5,
using a construction of Evans [27]. This examples will in fact be supersimple,
and $\omega$-categorical. We solve the problem in Section 5.3 for the original
setting of $\omega$-stable finite rank types using a more involved
construction involving $\aleph_{1}$-categorical $(r,k)$-Steiner systems. For
any $n\in\mathbb{N}$, we build rank $2$ types whose degree of nonminimality is
$n$.

Of course, many variations on the degree of nonminimality are possible, and we
introduce several new notions in Section 2. For instance, $F_{ind}(p)$ is
defined (in a simple theory) to be the least $n$ such that there exists an
infinite (nonconstant) indiscernible sequence $a_{i}$ in $p$, so that
$\operatorname{tp}(a_{n+1}/a_{\leq n})$ is a forking extension of $p$.333See
Definition 2.1.4. As we note - it isn’t hard to see that $F_{ind}(p)$ bounds
the degree of nonminimality, but the two invariants can be different, even in
the finite rank context, as we show in Section 5, though they are the same in
a large class of theories, including, for instance differentially closed
fields and compact complex manifolds. It is, however, interesting to note that
$F_{ind}(p)$ can be bounded in terms of the degree of nonminimality and the
Lascar rank of $p.$

In defining $F_{ind}(p)$ we quantify existentially over all indiscernible
sequences in the type $p$. In a definition which evokes Kim’s lemma, we
consider the same definition but replacing the existential quantifier with a
universal quantifier, defining $F_{Mb}\left(p\right)$ in Definition 2.1.6. In
special cases of the rank context, the notion is related to _pseudolinearity_
as considered for instance by Tomašić and Wagner [86]. This invariant and the
ideas around it play an essential role in Section 6.

###  1.2. Countable categoricity and the Koponen conjecture

There is a long and rich tradition of studying the intersection of countable
categoricity with tameness notions coming from classification theory. Before
explaining the particular setting of Koponen’s conjecture, which we resolve in
Section 6, we explain how the conjecture fits into this tradition. Lachlan
[61] conjectured that if $T$ is stable and $\omega$-categorical, then T is
totally transcendental and proves the weaker statement that if $T$ is
superstable and $\omega$-categorical, then T is totally transcendental.
Cherlin, Harrington, and Lachlan [20] later proved that superstable
$\omega$-categorical theories have finite Morley rank. Buechler [17] gave a
complete classification of $\omega$-stable $\omega$-categorical theories in
terms of a simple collection of invariants. Hrushovski [44] proved that any
totally categorical theory can be finitely axiomatized relative to the
collection of axioms saying the structure is infinite, completing the
resolution of questions about finite axiomatizability of categorical theories
pondered since the early 1960’s [66].

More recently, Palacín [74] showed that an $\omega$-categorical $CM$-trivial
simple theory is low and asked [74, see the Question on Page 10] if every
simple $\omega$-categorical theory is low.444This also appears as [18,
Question 2, page 755]. Palacín hints that without some additional assumption,
there isn’t strong evidence that that the question should have an affirmative
answer. Simon [83] classified $\omega$-categorical NIP structures of rank one.
A version of Simon’s classification result was extended to the higher rank
setting by Onshous and Simon [70]. Shelah [80] famously showed that any
unstable NIP structure is SOP, and asked if one can show the stronger
statement that the structure has an interpretable infinite linear order. Simon
[82] showed this in the case that the theory is assumed to be
$\omega$-categorical. There is an equally extensive literature around
$\omega$-categoricity in more specific algebraic contexts. See page 7 of [19]
for an extensive collection of references.

Koponen asks the following question [56], [58], [59], conjecturing a positive
answer to the question of supersimplicity of finite rank:

######  Question 1.2.1.

Is every simple theory with quantifier elimination in a finite relational
language one-based (in particular, supersimple of finite rank)?

For a theory to have quantifier elimination in a finite relational language is
the same thing as being $\omega$-categorical and $n$-ary for some $n<\omega$
(that is, having quantifier elimination in language with only $n$-ary relation
symbols). Assuming $T$ as in the question is not just simple, but supersimple,
we will see in Section 6 a positive answer to Question 1.2.1 follows with a
bit of effort from a result of Tomašić and Wagner [86], appearing in this
paper as Fact 3.2.11. Again in the supersimple case, we give a second proof
which goes through our development of $F_{Mb}$ via Corollary 3.2.20. This uses
the forking degrees $F_{\lambda}$ (Definition 3.2.18), originally introduced
in [30] to generalize the degree of nonminimality. Either proof supplies us
with invariants that distinguish infinitely many indiscernible sequences over
a finite set, thereby contradicting the assumption of quantifier elimination
in a finite language.

In any case, the most difficult part of our proof of the full Koponen
conjecture involves showing that a theory which is simple with quantifier
elimination in a finite relational language must be supersimple. In less
syntactic terms, the complicated part is showing that every
$\omega$-categorical $n$-ary theory which is simple is in fact supersimple. In
the finite rank case, the result of [86] produces canonical bases (for types
with fixed variables) of arbitrarily large $\mathrm{SU}$-rank–or equivalently,
in the sense of Corollary 3.2.4 and Corollary 3.2.9, a type $q$ with
$F_{Mb}(q)=\infty$. In the general supersimple case, [86] produce canonical
bases of arbitrarily large finite $\mathrm{SU}_{p}$-rank, where $p$ is a
regular type, again producing types $q$ with $F_{Mb}(q)=\infty$. So in the
supersimple case, we are able to distinguish infinitely many indiscernible
sequences over a finite set according to the $\mathrm{SU}$\- or
$\mathrm{SU}_{(p)}$-ranks of the canonical bases of their limit types.
However, in the simple but not supersimple setting, $\mathrm{SU}$-rank will
not be well-defined, nor will we necessarily get nontrivial ranks of the form
$\mathrm{SU}_{p}$. To remedy this, in Definition 3.2.15 we introduce a new
notion of rank for general simple theories, the $\mathrm{G}$-rank of a set.
Then in Theorem 6.3, we show that, in strictly simple $\omega$-categorical
theories, there is an invariant distinguishing infinitely many indiscernible
sequences (with terms of fixed length), given by producing canonical bases of
arbitrarily large finite $\mathrm{G}$-rank.

Previously, partial progress on the Koponen conjecture (from each of the
different perspectives described above) can be found in numerous sources [3,
1, 56, 57, 75, 58]. Aranda [3] shows that binary homogeneous relational
structures can not have infinite monomial $SU$-rank. Ahlman and Koponen [1]
develop restrictions (towards the structure being binary) on the minimal (that
is, $SU$-rank $1$) definable sets in $\omega$-categorical simple structures in
a finite relational language. Koponen [56] resolved the conjecture when the
theory is additionally assumed to be binary. Palacin [75] proves Koponen’s
conjecture in special cases with additional assumptions (e.g. the theory is
binary with $2$-transitive automorphism group). Koponen [58] solves the
special case when the relations are ternary. However, outside of the binary
and ternary cases, little was previously known, to our knowledge, about the
Koponen conjecture. Following our proof of Koponen’s conjecture, we formulate
a generalization to $\mathrm{NSOP}_{1}$ theories.

###  1.3. Stable forking and generalizations

The ideas investigated in Section 7 have their origins in the _stable forking
conjecture_ , that every instance of forking in a simple theory is witnessed
by a stable formula [53]. In previous sections, we introduced a number of
invariants which allowed us to quantitatively limit the complexity of forking.
In Section 7, we relate these measures to older qualitative limits and
characterizations of forking around the stable forking conjecture including,
for instance, the variants considered Palacín and Wagner [75]. Bounding
$F_{Mb}\left(p\right)$ is one of several quantitative ways in this paper in
which we’ve bounded the complexity of forking in various specific contexts.
When $F_{Mb}\left(p\right)$ is finite, one might ask for a further reduction
in the complexity of forking - among all indiscernible sequences in type $p$,
is the collection of Morley sequences relatively definable? This holds, for
example, in the setting of the Koponen conjecture, when a theory is assumed to
be $\omega$-categorical and $n$-ary. We formalize this _definable Morley
property_ of indiscernible sequences in Definition 7.18 and investigate its
properties, showing in Theorem 7.20 that it gives us nontrivial instances of
the simple Kim-forking conjecture (Question 7.3) in $\mathrm{NSOP}_{1}$
theories. As a special case, we get nontrivial instances of the simple Kim-
forking conjecture for types $p$ over finite sets with $F_{Mb}(p)<\infty$,
assuming a type-counting condition.

In a slightly different direction, Palacín and Wagner [75] investigated the
forking relation (not a priori a definable relation) itself as a ternary
relation - asking in a variety of circumstances if this relation itself is
stable. For instance, Wagner and Palacín show that in a supersimple CM-trivial
theory, the independence relation is stable, that is,
$x\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{y_{1}}y_{2}$ can not order an infinite
indiscernible sequence. In Section 7, we expand this investigation, broadening
and formalizing what it means to ask if the (Kim)-forking relation is simple.
As a consequence of Kaplan and Ramsey’s results ([49], stated here as Fact
7.9) we show in Theorem 7.10 that with enough variables, this global analogue
of the simple Kim-forking conjecture holds over a model in any
$\mathrm{NSOP}_{1}$ theory. In the setting of where $F_{Mb}(p)<\infty$, we
show in Theorem 7.14 that only finitely many variables are required for this
global version of the simple Kim-forking conjecture. As we shall also see, in
the $\omega$-categorical setting, some of the boundaries between these
properties and our earlier notions dissolve.

###  1.4. Overall Organization of the paper

In Section 2, we introduce various invariants, and prove some basic facts
about the notions. In Section 3 we give examples of some nontrivial
computations of the invariants introduced in the previous section. We also
establish some more involved inequalities. Section 4 centers around the
invariant $F_{Mb}\left(p\right)$, introduced in Section 2, relating it to
existing work on psudolinearity and giving several nontrivial examples. We
also prove some results about the robustness of the invariant. For instance,
we show it is preserved under equidomination of types. We use the theory of
free projective planes of Hyttinen and Paolini [46] to show that in a stable
theory, $F_{Mb}\left(p\right)$ can take on any positive integer value. Section
5 is devoted to answering the problems of [33, 35], showing that the degree of
nonminimality can take on any positive integer value. Section 6 resolves
Koponen’s conjecture. Section 7 introduces new variations of the stable
forking conjecture which go beyond simple theories. Several of these
variations naturally arise from the study of the invariant
$F_{Mb}\left(p\right)$.

##  2\. Degrees, nonminimality, and invariants of indiscernibility

In this section we define several ranks for the complexity of forking and
compare them.

###  2.1. Degree of nonminimality

In [35], the following notion was defined:

######  Definition 2.1.1.

(Degree of nonminimality of a stationary type in a stable theory) Given a
stationary type $p\in S(A)$, in a stable theory $T$ with $\mathrm{U}(p)>1$,
the degree of nonminimality of $p$ is the minimal length $n$ of a sequence
$a_{1},\ldots,a_{n}$ of realizations of the type $p$ such that $p$ has a
nonalgebraic forking extension over $a_{1},\ldots,a_{n}$.

The same definition makes sense in the case of a general type in an arbitrary
theory (though it seems very little could be proved in that case!). We will
essentially only develop the notion in the settings where $T$ is a simple
theory (or later, an $\mathrm{NSOP}_{1}$ theory):

######  Definition 2.1.2.

Let $T$ be simple. Given a type $p\in S(A)$ the degree of nonminimality of $p$
($\operatorname{nmdeg}(p)$) is the minimal length $n$ of a sequence of
realizations of the type $p$, say $a_{1},\ldots,a_{n}$ such that $p$ has a
nonalgebraic forking extension over $Aa_{1},\ldots a_{n}$. If there is no such
$n$ then we let $\operatorname{nmdeg}(p)=\infty.$ If $p$ is a minimal type and
thus has no nonalgebraic forking extension, then we set
$\operatorname{nmdeg}(p)=1$ as a matter of convention.

######  Remark 2.1.3.

To calculate $\operatorname{nmdeg}(p)$, it is sufficient to consider
$\bar{a}=a_{1},\ldots,a_{n}$ to be independent over $A$. To see this, note
that if $a_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}\bar{a}_{<i}$, then if $n$ is
$\operatorname{nmdeg}(p)$ and $i<n$, it must be that
$a_{i}\in\operatorname{acl}(A\bar{a}_{<i}).$ In this case, if $p$ has a
nonalgebraic forking extension over $\bar{a}$, then $p$ has a nonalgebraic
forking extension over
$\bar{a}_{-i}=(a_{1},\ldots,a_{i-1},a_{i+1},\ldots,a_{n})$. But this is
impossible if $n=\operatorname{nmdeg}(p)$. So, it must be that $\bar{a}$ is an
independent tuple of realizations of $p$.

Next, we consider a small variation on the idea of the degree of
nonminimality.

######  Definition 2.1.4.

Let $T$ be simple, and $p\in S(A)$. Let $F_{ind}(p)$ be the least
$n\in\mathbb{N}$ such that there is a nonconstant $A$-indiscernible sequence
$(a_{i})_{i\in\omega}$ in the type $p$ such that
$\operatorname{tp}(a_{n}/Aa_{<n})$ is a forking extension of $p$. If no such
$n$ exists, then define $F_{ind}(p)=\infty$. When $p$ is minimal, any
nonconstant indiscernible sequence is a Morley sequence, and we set
$F_{ind}(p)=1$ as a matter of convention.

######  Remark 2.1.5.

Note that in Definition 2.1.4, since the sequence $(a_{i})$ is indiscernible
and nonconstant, $\operatorname{tp}(a_{n+1}/a_{\leq n})$ must be a
nonalgebraic forking extension of $p$. Because of this, one can see
immediately that $F_{ind}(p)\geq\operatorname{nmdeg}(p).$ Moreover, if
$\mathrm{SU}(p)>1$, for $p\in S(A)$, $F_{ind}(p)$ is finite: let $q\in S(B)$
be a nonalgebraic forking extension, and let $(a_{i})_{i\in\omega}$ be a
Morley sequence over $B$ where for $i<\omega$, $a_{i}\models q$. Then
$(a_{i})_{i<\omega}$ is not constant, but is not a Morley sequence over $A$:
if it were, by Kim’s lemma, $B\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{0}$, but by symmetry,
$a_{0}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}B$, contradicting that $q$ is a forking
extension of $p$. So $\operatorname{tp}(a_{n+1}/a_{\leq n})$ is a forking
extension of $p$ for some $n$, and $F_{ind}(p)\leq n$. In Section 3 we will
develop the properties of $F_{ind}(-).$

In the original primary applications of $\operatorname{nmdeg}(-)$
(differentially closed fields and compact complex manifolds), we will see in
Proposition 3.1.3 that the two quantities are identical. In Section 5 we will
show that the two quantities are not equal in general, giving examples where
each is a different finite number. It is, however, interesting to note that in
Section 3.2 we will show the two quantities are bounded in terms of each
other.

So, the quantity $F_{ind}(p)$ is the least $n$ so that _there exists_ an
indiscernible sequence $(a_{i})$ in the type $p$, for which we can see that
$(a_{i})$ is not a Morley sequence by looking only at
$a_{1},a_{2},\ldots,a_{n+1}$. By changing the _existential_ to a _universal
quantifier_ over the collection of indiscernible sequences in $p$, we arrive
at the following definition:

######  Definition 2.1.6.

Let $T$ be simple. Let $p\in S(A)$ be nonalgebraic. Let $F_{Mb}(p)$ be the
least $n\in\mathbb{N}$ such that for all $A$-indiscernible sequences
$(a_{i})_{i\in\omega}$ in the type $p$ that are not Morley sequences,
$\operatorname{tp}(a_{n}/Aa_{<n})$ is a forking extension of $p$. If no such
$n$ exists, then define $F_{Mb}(p)=\infty.$

Note that when $\mathrm{SU}(p)>1$, because there is, as in Remark 2.1.5, a
nonconstant $A$-indiscernible sequence in $p$ that is not a Morley sequence
over $A$, we can take Definition 2.1.6 to quantify over all nonconstant
$A$-indiscernible sequences that are not Morley sequences, as in Definition
2.1.4. We may also state the relationship between $F_{ind}(p)$ to $F_{Mb}(p)$
as follows: $F_{ind}(p)$ is the least $n$ so that there is a nonconstant
$A$-indiscernible sequence $\\{a_{i}\\}$ in $p$ that is $n$-independent over
$A$ (i.e. any $n$ of the $a_{i}$ are independent over $A$) but not
$n+1$-independent over $A$, while $F_{Mb}(p)$ is the greatest $n$ so that
there is an $A$-indiscernible sequence $\\{a_{i}\\}$ in $p$ that is
$n$-independent over $A$ but not $n+1$-independent over $A$.

We first compute $F_{Mb}(p)$ in the most basic case:

######  Definition 2.1.7.

Suppose the type $p\in S(A)$ has the property that for all $a\models p$, if
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}B$ then
$\mathrm{acl}^{eq}(aA)\cap\mathrm{acl}^{eq}(BA)\supsetneq\mathrm{acl}^{eq}(A)$.
In this case, let us say that $p$ has _algebraic forking_.

This generalizes one-basedness in the following sense:

######  Fact 2.1.8.

(see, e.g. [77] or [87]) A simple theory with elimination of hyperimaginaries
is one-based if and only if every type has algebraic forking.

In general, whenever
$\mathrm{acl}^{eq}(aA)\cap\mathrm{acl}^{eq}(BA)\supsetneq\mathrm{acl}^{eq}(A)$,
we have $a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}B$, so:

######  Proposition 2.1.9.

If the type $p\in S(A)$ has algebraic forking and $a\models p$, then

$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}B\iff\mathrm{acl}^{eq}(aA)\cap\mathrm{acl}^{eq}(BA)\supsetneq\mathrm{acl}^{eq}(A).$

For example, if $SU(p)=1$ or $p$ is one-based, then $p$ has algebraic forking.

######  Proposition 2.1.10.

Suppose $p$ has algebraic forking. Then $F_{Mb}(p)=1$.

###### Proof.

Suppose $(a_{i})_{i<\omega}$ is an $A$-indiscernible sequence where
$a_{0}\models p$, and is not a Morley sequence over $A$. If $p$ has algebraic
forking, then for some $n<\omega$,
$\mathrm{acl}(a_{n}A)\cap\mathrm{acl}(a_{0}\ldots a_{n-1}A)=B\supsetneq A$.
The sequence $(a_{n+i})_{i<\omega}$ is indiscernible over
$\mathrm{acl}(a_{0}\ldots a_{n-1}A)$, so in particular, is indiscernible over
$B$. Then $B\subseteq\mathrm{acl}(a_{n}A)$, so
$B\subseteq\mathrm{acl}(a_{n+1}A)$, and
$\mathrm{acl}(a_{n}A)\cap\mathrm{acl}(a_{n+1}A)\supseteq B\supsetneq A$. So
$a_{n+1}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{n}$, and by indiscernibility,
$a_{1}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{0}$. ∎

Excepting for edge cases in which $\operatorname{nmdeg}(p)$ either doesn’t
make sense or is a matter of convention (e.g. when $\mathrm{U}(p)=1$), the
definitions and above discussion make it transparent that
$\operatorname{nmdeg}(p)\leq F_{ind}(p)\leq F_{Mb}\left(p\right)$, but as
we’ll show in Section 3, the quantities can all be distinct.

###  2.2. Related notions involving canonical bases

In this subsection, in order to formulate the notions, we must work in a
setting with a suitable notion of canonical bases for types. For instance, it
would suffice to work in a stable theory with elimination of imaginaries (or
else in our formulations, work in $T^{eq}$). More generally, one might work in
a simple theory with elimination of hyperimaginaries.

Each of the three notions given above in Definitions 2.1.2, 2.1.4, 2.1.6 can
be related to the rank of the canonical base of certain forking extensions of
the type $p$; we will explain the connection in the coming pages.

######  Definition 2.2.1.

Given a type $p,$ let

  1. (1)

$F_{cb}\left(p\right)$ be the minimal number $m$ of realizations $a_{1},\ldots
a_{m}$ of $p$ such that the canonical base of $p$ is contained in the
algebraic closure of $a_{1},\ldots,a_{m}$.

  2. (2)

$F_{Icb}\left(p\right)$ be the minimal number $m$ of _independent_
realizations $a_{1},\ldots a_{m}$ of $p$ such that the canonical base of $p$
is contained in the algebraic closure of $a_{1},\ldots,a_{m}$.

Though it isn’t obvious, the quantity $F_{Icb}\left(p\right)$ is intensively
studied in the context of differential equations and modeling, where the
notion is closely related to parameter identifiability, as we will explain. In
parameter identifiability problems, one typically considers variations on the
following question: given a fixed system of differential equations with
unknown parameters and some output functions, can one identify the unknown
parameters from a series of observations of independent solutions to the
system? Numerous variations on this general idea are extensively studied in
the differential equations literature. See for instance [25, 71, 47]

Specifically, when $p$ is the generic type of the solution set of a system of
differential equations, $\Sigma$, the notion $F_{Icb}\left(p\right)$ is
closely related to the minimum value of $r$ such that the parameters of
$\Sigma$ are multi-experiment identifiable in $r$ many experiments [72,
Section 2.2]. There is a slight difference in that we are considering the
algebraic closure, while in parameter identifiability problems, the definable
closure is used. This is a very small difference, and in the setting of
parameter identifiability would yield values of $r$ which differ by at most
one. In the differential context, the emphasis has often been on the design of
efficient algorithms for checking the identifiability of a system.

It is natural to define the relative versions of these notions for forking
extensions:

######  Definition 2.2.2.

Given a type $p\in S(\emptyset)$ and $q$ a forking extension of $p$ let:

  1. (1)

$F_{cb}\left(q/p\right)$ be the minimal number $m$ of realizations
$a_{1},\ldots a_{m}$ of $p$ such that the canonical base of $q$ is contained
in the algebraic closure of $a_{1},\ldots,a_{m}$

  2. (2)

$F_{Icb}\left(q/p\right)$ be the minimal number $m$ of independent
realizations $a_{1},\ldots a_{m}$ of $p$ such that the canonical base of $q$
is contained in the algebraic closure of $a_{1},\ldots,a_{m}$.

Given a type $p$ of rank larger than one, there is a type $q$ such that
$\operatorname{nmdeg}(p)=F_{cb}\left(q/p\right)$; in fact:

$\displaystyle\operatorname{nmdeg}(p)$ $\displaystyle=$
$\displaystyle\min\left\\{F_{cb}\left(q/p\right)\,|\,q\text{ a nonalgebraic
forking extension of }p\right\\}$ $\displaystyle=$
$\displaystyle\min\left\\{F_{Icb}\left(q/p\right)\,|\,q\text{ a nonalgebraic
forking extension of }p\right\\}.$

##  3\. Basic properties of the invariants

###  3.1. Basic Examples

We next give a series of examples centered around the invariants of the
previous section.

######  Example 3.1.1.

In any strongly minimal theory, if $a,b,c$ are independent the generic
$2$-type $p$ has $\operatorname{nmdeg}(p)=F_{ind}(p)=1$, witnessed by $(a,b)$
and $(a,c)$. Similarly, for any $n$-type of rank at least $2$ in a strongly
minimal theory.

######  Example 3.1.2.

For $2$-types in strongly minimal theories, the analysis of $F_{Mb}$ is more
interesting. For instance, as we will explain, in the theory of algebraically
closed fields, the generic $2$-type has $F_{Mb}(p)=\infty$.

Let $T=ACF_{0}$, the theory of algebraically closed fields of characteristic
zero. Let $p\in S_{2}(\emptyset)$ be the generic type of affine $2$-space.
Consider the indiscernible sequence in $p$ given by independent generic points
of a generic affine curve of degree $d$,

$\sum_{i,j}c_{i,j}x^{i}y^{j}+1=0,$

where $c_{i,j}$ are independent transcendentals. Let $q$ be the generic type
of the curve. It isn’t hard to see that the tuple $\bar{c}$ of the
transcendentals $c_{i,j}$ is the canonical parameter of the forking extension
$q$ of $p$, and that $\bar{c}$ is in the field generated by $(d+1)^{2}-1$ many
independent realizations of $q$, but no fewer (each point yields one
independent linear condition on the coefficients). So, taking an indiscernible
sequence in $q$, $(a_{i})$, over $\emptyset$, $(a_{i})_{i\leq(d+1)^{2}-1}$ is
an initial segment of a Morley sequence (ie this collection of points in
independent), but $(a_{i})_{i\leq(d+1)^{2}}$ is not. Since a similar example
exists for any $d\in\mathbb{N}$, we can see that $F_{Mb}{(p)}=\infty$.

The above example, while familiar and illustrative, turns out to not be
particular to $ACF_{0}$, strongly minimal types, or even types in stable
theories. In general, in Corollary 3.2.12, we show that if $p$ is a Lascar
type with $\mathrm{SU}(p)=1$ (in a stable or $\omega$-categorical simple
theory) then either $F_{Mb}\left(p\otimes p\right)=\infty$ or $p$ is
_linear_.555See Definition 3.2.7. In Section 5.2 we give an example of a type
of $\mathrm{SU}$-rank $2$ in an $\omega$-categorical supersimple theory of
finite rank in which $\operatorname{nmdeg}(p)<F_{ind}(p)$, but this is rather
more involved than the basic examples of this section. Examaples of stable
theories with $\operatorname{nmdeg}(p)<F_{ind}(p)$ are given in Section 5. In
many cases, the two invariants are identical - see remarks following the proof
of our next result.

Let $DCF_{0}$ denote the theory of differentially closed fields of
characteristic zero in the language of rings together with a distinguished
derivation $\delta$. The proof of the following result is quite similar to
that found in Section 2 of [33].

######  Proposition 3.1.3.

For a type $p$ in the theory $DCF_{0}$, $\operatorname{nmdeg}(p)=F_{ind}(p).$

###### Proof.

Let $p\in S(A)$. If $p_{n}\rightarrow p_{n-1}\rightarrow\ldots\rightarrow
p_{1}$ is an analysis of the type $p$ over $A$, $F_{ind}(p)$ is bounded by
$F_{ind}(p_{1})$ and also $F_{ind}$ of the types given by the generic fibers
of the maps in the analysis. Thus, it suffices to prove the result in the case
that $p$ has _no proper fibrations_ [65, Definition 2.1]. It follows by [65,
Proposition 2.3] that $p$ is either

  1. (1)

interalgebraic over $A$ with a finite collection of independent realizations
of a locally modular minimal type, or

  2. (2)

interalgebraic with a type $q$ which is internal to a nonlocally modular type.

In the first case, both $F_{ind}(p)=\operatorname{nmdeg}{(p)}=1$. In the
second case, if the binding group of $p$ relative to the constants is not
generically $2$-transitive [35, for the defintion of generic transitivity],
then $F_{ind}(p)=1$, and $\operatorname{nmdeg}(p)\leq F_{ind}(p)$ in general
(see Remark 2.1.5), so the quantities are equal. In the second case, since
$F_{ind}(-)$ is invariant up to interalgebraicity, we can assume $p$ is
internal to the nonlocally modular type - in this case, the constants.

Suppose now that $F_{ind}(p)>1$. Then it must be that $p$ and also $p\otimes
p$ are weakly orthogonal to the constants over $A$ (if not, it is easy to see
that $F_{ind}(p)=1$ by taking a Morley sequence whose image in
$\mathcal{C}^{n}$ coming from nonorthogonality lies on an algebraic curve). It
follows that both $p$ and $p\otimes p$ are isolated types. We wish now to
reduce to the case that the action of the binding group $G$ on $p$ is
$2$-transitive. This is equivalent to any two realizations of $p$ being
independent. As the binding group is generically $2$-transitive, with
$p\otimes p$ being isolated, there must be an $A$-definable equivalence
relation $\sim$ on $p$ such that $p/\sim$ has a $2$-transitive binding group.

But now the binding group is $2$-transitive and definably isomorphic to an
algebraic group action acting on an algebraic variety over a field of
characteristic zero, and so the classification theorem of Knopp [54] tells us
that the binding group action on $p$ is isomorphic to either a group of affine
transformations of affine space or the action of $PGL_{n+1}$ on
$\mathbb{P}^{n}.$ Here it must be that $n>1.$ Both of these group actions
preserve collinearity of points. Taking the Morley sequence of points on a
line, we can see that $F_{ind}(p)=2.$ $\operatorname{nmdeg}(p)=2$ in this case
as well since the binding group acts $2$-transitively. ∎

Again, similar to the results of Section 2 of [33], in the previous result,
one does not really need to assume that the theory $T$ is $DCF_{0}$. Rather,
it applies more generally when $T$ is $\omega$-stable, eliminates
$\exists^{\infty}$, and any nonlocally modular minimal type over $A$ is
nonorthogonal to a pure algebraically closed field over $A$. So, for instance,
the proof applies to the theory of compact complex manifolds or more generally
the class of theories laid out in Section 3 of [33].

The proof of the previous result also gives a reasonable indication of the
difficulties associated with giving examples in which
$\operatorname{nmdeg}(p)\neq F_{ind}(p)$ \- one either has to give an example
of a theory whose binding groups have different transitivity properties than
algebraic groups or give a theory in which any minimal type in some
nonorthogonality class of minimal types requires parameters to define. We
don’t really know examples of such groups - indeed conjectures like those of
Cherlin-Zilber, Borovik-Cherlin [13], and Borovik-Deloro [14] are towards the
nonexistence of such groups. Most natural theories don’t seem to have the
second property, but we build such a (simple) theory in Section 5.2 using a
construction of Evans, and show that for our example there are types $p$ with
$\operatorname{nmdeg}(p)\neq F_{ind}(p)$.

######  Example 3.1.4.

Work in the theory $DCF_{0}$, and let $p$ be the generic type of the equation
$x^{\prime\prime}=0.$ It isn’t hard to see that over $\mathbb{Q}(t)$ the
solutions of $x^{\prime\prime}=0$ are in bijective correspondence with
$\mathbb{A}^{2}(\mathbb{C})$ with the induced structure of definable subsets
definable in $ACF_{0}$ (e.g. by stable embeddedness). Let $q$ be the forking
extension of solutions of the form $at+b$ where $(a,b)$ is a generic point on
the curve $a=\prod_{i=1}^{2d}(b-c_{i})$ where $c_{i}$ are independent
transcendental constants. Then taking solutions of $q$, one can see that $2d$
many are required to define the canonical base of $q$ (each solution gives
another point on the curve $a=\prod_{i=1}^{2d}(b-c_{i})$ and imposes one
linear condition on the coeficients $c_{i}$). On the other hand, taking
realizations of $p$ given by $c_{i}t+c_{2d-i}$, it is easy to see that a
canonical base of $q$ is contained in the $d$ many realizations as $i$ ranges
from $1$ to $d$. In this example we have:

$F_{cb}\left(q/p\right)=F_{Icb}\left(q/p\right)<F_{cb}\left(q\right)=F_{Icb}\left(q\right).$

Especially with regard to differential algebra, the quantities
$F_{cb}\left(q/p\right),\,F_{Icb}\left(q/p\right),\,F_{cb}\left(q\right),\,F_{Icb}\left(q\right)$
have natural interpretations in terms of existing work. For instance, when $q$
is the generic type of a solution to a system of ODE, and the parameters of
the system are unknown constants which form a canonical base,
$F_{Icb}\left(q\right)$ or $F_{Icb}\left(p\right)+1$ is the number of
experiments required to identify the parameters. In general,

$\operatorname{nmdeg}{p}=\min\left\\{F_{cb}\left(q/p\right)\,|\,q\text{ is a
nonalgebraic forking extension of }p\right\\}$

###  3.2. Inequalities and basic results

In this section, we assume that $T$ is simple. We will see that in a simple
theory, $\mathrm{nmdeg}(p)$ can be strictly less than $F_{ind}(p)$.
Nonetheless, in the finite-rank setting, the two quantities are related (can
be bounded in terms of each other). The inspiration for these arguments comes
from problems of parameter identifiability of differential equations; thanks
to Scanlon and Pillay for discussions around this topic at various times.

######  Lemma 3.2.1.

Let $T$ be a simple theory, and666$c\in\mathrm{bdd}(A)$ means $c$ has
boundedly many images over A under automorphisms of $M$; for the more
intricate definition of $S(\mathrm{bdd}(A))$, see [39, Definition 1.2]. $q\in
S(\mathrm{bdd}(A))$. Let $\\{a_{i}\\}_{i<\omega}$ be a Morley sequence in $q$
over $A$. Suppose $\mathrm{SU}(\mathrm{Cb}(q))=n<\omega$. Then
$\mathrm{Cb}(q)\subseteq\mathrm{bdd}(\\{a_{i}\\}_{i<n})$.

###### Proof.

Let $C=\mathrm{Cb}(q)$, and suppose otherwise; we show that, for $0\leq i\leq
n$, $a_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{a_{<i}}C$, contradicting
$\mathrm{SU}(\mathrm{Cb}(q))=n$. We may assume $A=\mathrm{bdd}(A)$, so
$C\subseteq A$. Suppose that for some $0\leq i\leq n$,
$a_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{a_{<i}}C$. Since $a_{i}\models q$,
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}A$. Since $\\{a_{i}\\}_{i<\omega}$ is a
Morley sequence in $p$ over $A$, $a_{i}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{A}a_{<i}$. So
by transitivity, $a_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}Aa_{<{i}}$, and therefore
$a_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{Ca_{<i}}Aa_{<{i}}$. By the assumption and
transitivity, $a_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{a_{<i}}Aa_{<_{i}}$. Since
$a_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{<i}$,
$C=\mathrm{Cb}(q)\subseteq\mathrm{bdd}(\mathrm{Cb}(\mathrm{tp}(a_{i}/Aa_{<i})))$.
But by $a_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{a_{<i}}Aa_{<_{i}}$,
$\mathrm{Cb}(\mathrm{tp}(a_{i}/Aa_{<i}))\subseteq\mathrm{bdd}(a_{<i})$, so
$C\subseteq\mathrm{bdd}(a_{<i})\subseteq\mathrm{bdd}(a_{<n})$, a
contradiction. ∎

This allows us to bound $F_{ind}(p)$ in terms of $\mathrm{nmdeg}(p)$, when $p$
is of finite rank.

######  Corollary 3.2.2.

Let $T$ be simple and $\mathrm{SU}(p)=n<\omega$. Then $\mathrm{nmdeg}(p)\leq
F_{ind}(p)\leq n\cdot\mathrm{nmdeg}(p)$.

###### Proof.

Assume without loss of generality that $p\in S(\emptyset)$. For the remaining
inequality (by Remark 2.1.5, $\operatorname{nmdeg}(p)\leq F_{ind}(p)$), let
$\mathrm{nmdeg}(p)=N$. Then there are $a_{1},\ldots,a_{N}\models p$ so that
$p$ has a nonalgebraic forking extension $q$ over $\mathrm{bdd}(a_{1}\ldots
a_{N})$. Let $I=\\{a^{\prime}_{i}\\}_{i<\omega}$ be a Morley sequence in $q$
over $\mathrm{bdd}(a_{1}\ldots a_{N})$; then $I$ is nonconstant because $q$ is
nonalgebraic. Because $\mathrm{Cb}(q)\subseteq\mathrm{bdd}(a_{1}\ldots
a_{N})$, by the Lascar inequalities, $U(\mathrm{Cb}(q))\leq n\cdot N$. So by
the previous proposition,
$C=\mathrm{Cb}(q)\subseteq\mathrm{bdd}(a^{\prime}_{<Nn})$. But
$a_{Nn}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{<Nn}$ because $I$ is a Morley sequence
over $A$, and $a_{Nn}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}A$ because $a\models q$, so
$a_{Nn}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}Aa_{<Nn}$ by transitivity, and therefore
$a_{Nn}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{a_{<Nn}}Aa_{<Nn}$ because
$C\subseteq\mathrm{bdd}(a^{\prime}_{<Nn})$. Thus
$a_{Nn}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{a_{<Nn}}A$. But because $a_{Nn}\models q$ and
$q$ forks over $p$, $a_{Nn}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}A$, so $a_{Nn}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}a_{<Nn}$, showing $F_{ind}(p)\leq nN$. ∎

In the finite-rank case (or the finite $\mathrm{SU}_{p_{0}}$-rank case, where
$p_{0}$ is a regular type), the ranks of canonical bases will bound how much
independence an indiscernible sequence has. In particular, each indiscernible
sequence will be a Morley sequence in a type over some set, and the canonical
base of that type will give upper and, in some cases, lower bounds on $k$ so
that this sequence is $k$-dependent if it is not a Morley sequence.

######  Proposition 3.2.3.

Let $A\subseteq B$ with $B=\mathrm{bdd}(B)$, $q\in S(B)$, $p=q|_{A}$. Let
$I=\\{a_{i}\\}_{i<\omega}$ be a Morley sequence in $q$ over $B$.

(1) If $\mathrm{SU}(\mathrm{Cb}(q)/A)\leq k$, where $1\leq k<\omega$, then
$a_{k}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{<k}$; that is, $I$ is $k+1$-dependent.

(2) If $\mathrm{SU}(q)=n<\omega$, $\mathrm{SU}(p)=n+1$, and
$\mathrm{SU}(\mathrm{Cb}(q)/A)>k$ where $1\leq k<\omega$, then
$a_{k-1}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{<k-1}$; that is, $I$ is $k$-independent.

###### Proof.

(1). Since $a_{k}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{B}a_{<k}$,
$\mathrm{bdd}(\mathrm{Cb}(\mathrm{tp}(a/\mathrm{bdd}(Ba_{<k}))=\mathrm{bdd}(\mathrm{Cb}(q))\subseteq\mathrm{bdd}(Aa_{<k})$,
the inclusion by Proposition 3.2.1, so $a_{k}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{Aa_{<k}}Ba_{<k}$. Assume for a contradiction
that $a_{k}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{<k}$; then by transitivity,
$a_{k}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}B$. But $a_{k}\models q$ and $q$ is a
forking extension of $p$ (since $\mathrm{SU}(\mathrm{Cb}(q)/A)\geq 1$), a
contradiction.

(2) Suppose, for a contradiction, that $a_{k-1}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{<k-1}$. Then
$n=\mathrm{SU}(a_{k}/B)=\mathrm{SU}(a_{k}/Ba_{k-1})\leq\mathrm{SU}(a_{k}/Aa_{k-1})\leq
n$, the first two equalities by $a_{k}\models q$ and the fact that $I$ is a
Morley sequence, the last by the assumption. So
$a_{k}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{Aa_{<k}}Ba_{<k}$. Then
$\mathrm{bdd}(\mathrm{Cb}(q))=\mathrm{bdd}(\mathrm{Cb}(\mathrm{tp}(a/\mathrm{bdd}(Ba_{<k}))\subseteq\mathrm{bdd}(Aa_{<k})$,
so $\mathrm{SU}(\mathrm{Cb}(q)/a_{<k})=0$, and it suffices to show that, for
$0<i<k$,
$\mathrm{SU}(\mathrm{Cb}(q)/A{a_{<i}})-\mathrm{SU}(\mathrm{Cb}(q)/A{a_{<i+1}})\leq
1$. By the Lascar inequalities, the quantity
$\mathrm{SU}(\mathrm{Cb}(q)/A{a_{<i}})-\mathrm{SU}(\mathrm{Cb}(q)/A{a_{<i+1}})$
is equal to the quantity
$\mathrm{SU}(a_{i}/Aa_{<i})-\mathrm{SU}(a_{i}/Aa_{<i}\mathrm{Cb}(q))$. But
$n+1=\mathrm{SU}(a_{i}/B)=\mathrm{SU}(a_{i}/a_{<i}B)$ while
$n=\mathrm{SU}(a_{i}/A)$, so because $A\subseteq Aa_{<i}\subseteq
Aa_{<i}\mathrm{Cb}(q)\subseteq\mathrm{bdd}(a_{<i}B)$, it follows that
$\mathrm{SU}(\mathrm{Cb}(q)/A{a_{<i}})-\mathrm{SU}(\mathrm{Cb}(q)/A{a_{<i+1}})\leq
1$.

∎

Note the resemblance between part (2) of this proposition and the direction
((b) $\Rightarrow$ (a)) of Theorem 3.3 of Kim ([51]), and the resemblance, in
the case where $\mathrm{SU}(p)=2$, between part (1) of this proposition and
the direction ((a) $\Rightarrow$ (b)) of that theorem of Kim, whose proof, as
stated in [51], uses similar ideas to the proof of Theorem 3.6 of [78]. We
expect the above proofs to be similar to a proof of Theorem 3.3 of Kim ([51]).

When $\mathrm{SU}(p)=1$, $\mathrm{nmdeg}(p)=F_{ind}(p)=F_{Mb}(p)=1$. So
$\mathrm{SU}(p)=2$ is the first case in which $\mathrm{nmdeg}(p)\leq
F_{ind}(p)\leq F_{Mb}(p)$ can be greater than $1$. In this case, by the
previous proposition, the possible ranks of the canonical bases of the forking
extensions of $p$ will give both upper and lower bounds for $F_{ind}(p)$ and
$F_{Mb}(p)$.

######  Corollary 3.2.4.

Let $p\in S(A)$. Let

$N_{-}(p)=\mathrm{min}(\\{\mathrm{SU}(\mathrm{Cb}(q)/A):p\subset q\in
S(B),A\subseteq
B=\mathrm{bdd}(B),\mathrm{SU}(q)>0,\mathrm{SU}(\mathrm{Cb}(q)/A)\geq 1\\})$

and

$N_{+}(p)=\mathrm{max}(\\{\mathrm{SU}(\mathrm{Cb}(q)/A):p\subseteq q\in
S(B),A\subseteq B=\mathrm{bdd}(B)\\}.)$

(If not defined, define $N_{+}(p)=\infty$.)

Then

(1) $F_{ind}(p)\leq N_{-}(p)$, and if $\mathrm{SU}(p)=2$,
$N_{-}(p)=F_{ind}(p)$.

(2) $F_{Mb}(p)\leq N_{+}(p)$, and if $\mathrm{SU}(p)=2$, $N_{+}(p)=F_{Mb}(p)$.

###### Proof.

Note that $I$ is an indiscernible sequence over $A$ that is not a Morley
sequence over $A$, if and only if $I$ is a Morley sequence over $B$ in $q\in
S(B),A\subseteq B=\mathrm{bdd}(B)$, for $q\supset p$ some forking extension of
$p$ (i.e. $\mathrm{SU}(\mathrm{Cb}(q)/A)\geq 1$). If additionally
$\mathrm{SU}(p)=2$, then $I$ is a nonconstant indiscernible sequence over $A$
that is not a Morley sequence over $A$, if and only if $I$ is a Morley
sequence over $B$ in $q\in S(B),A\subseteq B=\mathrm{bdd}(B)$, with
$\mathrm{SU}(q)=1$–thereby satisfying the hypothesis of (2) of the previous
proposition. So the corollary follows from the previous proposition by the
definitions of $F_{ind}$ and $F_{Mb}$.

∎

Even if $p$ is only assumed to be of finite rank, rather than rank $2$, the
existence of some $N$ so that $F_{Mb}(q)\leq N$ for all extensions $q\supseteq
p$ is equivalent to there existing some $N^{\prime}$ so that all extensions
$q\supseteq p$ have the property that the canonical bases of forking
extensions of $q$ have rank at most $N$.

######  Proposition 3.2.5.

Let $p\in S(A)$, $\mathrm{SU}(p)<\omega$. Suppose there is some $N$ so that
for all $q\supseteq p$, $q\in S(B)$, $A\subseteq B$, $F_{Mb}(q)\leq N$. Then
there is some $N^{\prime}$ so that for all $q\supseteq p$, $q\in S(B)$,
$A\subseteq B$, $N_{+}(q)\leq N^{\prime}$ (where $N_{+}(q)$ is as in the
previous corollary). (So the converse also holds, by the clause $F_{Mb}(p)\leq
N_{+}(p)$ of part (2) of the previous corollary.)

###### Proof.

We claim that suffices to show the following: Suppose that $p$ has the
property that for $q\supseteq p$, $q\in S(B)$, $A\subseteq B$, $F_{Mb}(q)\leq
N$, and that $\mathrm{SU}(p)\leq k$. Then if $I=\\{a_{i}\\}_{i<\omega}$ is an
$A$-indiscernible sequence of realizations of $p$,
$a_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{Aa_{<kN}}a_{<i}$ for $i\geq kN$. To show this
suffices, assume we have shown this and let $r\supseteq p$, $r\in S(B)$,
$A\subseteq B=\mathrm{bdd}(B)$; we show that
$\mathrm{SU}(\mathrm{Cb}(r)/A)\leq k^{2}N$, so that $N_{+}(p)\leq k^{2}N$–this
is enough, since $\mathrm{SU}(q)\leq k$ for $q\supseteq p$ so we will also
have shown that for all $q\supseteq p$ $q\in S(B)$, $A\subseteq B$,
$N_{+}(q)\leq k^{2}N$. To show $\mathrm{SU}(\mathrm{Cb}(q)/A)\leq k^{2}N$, let
$I$ be a Morley sequence in $r$ over $B$, so that
$a_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{Aa_{<kN}}a_{<i}$ for $i\geq kN$ by
assumption. Choose $a_{\omega}$ so that $I\smallfrown\langle
a_{\omega}\rangle$ remains a Morley sequence over $B$; by strong finite
character, we also have that $a_{\omega}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{Aa_{<kN}}I$.
Then since $I\smallfrown\langle a_{\omega}\rangle$ is a Morley sequence over
$B$, $\mathrm{tp}(a_{\omega}/BI)$ is a nonforking extension of $q$, so
$\mathrm{bdd}(\mathrm{Cb}(\mathrm{tp}(a_{\omega}/BI)))=\mathrm{bdd}(\mathrm{Cb}(q))$,
but by finite satisfiability, $\mathrm{tp}(a_{\omega}/BI)$ is itself a
nonforking extension of $\mathrm{tp}(a_{\omega}/AI)$, so by
$a_{\omega}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{Aa_{<kN}}I$ and transitivity, of
$\mathrm{tp}(a_{\omega}/Aa_{<kN})$. So
$\mathrm{bdd}(\mathrm{Cb}(q))=\mathrm{bdd}(\mathrm{Cb}(\mathrm{tp}(a_{\omega}/BI)))\subseteq\mathrm{bdd}(Aa_{<kN})$,
so by the Lascar inequalities, $\mathrm{SU}(\mathrm{Cb}(q)/A)\leq k^{2}N$.
(Note that, using the ideas of the proof of part (2) of Proposition 3.2.3, we
could have improved this bound.)

Therefore, it remains to show that if $p$ has the property that for
$q\supseteq p$, $q\in S(B)$, $A\subseteq B$, $F_{Mb}(q)\leq N$, and that
$\mathrm{SU}(p)\leq k$, and if $I=\\{a_{i}\\}_{i<\omega}$ is an
$A$-indiscernible sequence of realizations of $p$, then
$a_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{Aa_{<kN}}a_{<i}$ for $i\geq kN$. We show this
by induction on $k$. If $p$ is already a Morley sequence, then we are done. If
not, then by $F_{Mb}(q)\leq N$, $p^{\prime}=\mathrm{tp}(a_{N}/Aa_{<N})$ is a
forking extension of $p$ and has $\mathrm{SU}(p)\leq k-1$;
$I=\\{a_{i}\\}_{N\leq i<\omega}$ is an $Aa_{<N}$-indiscernible sequence of
realizations of $p^{\prime}$. Moreover, $p^{\prime}$ continues to have the
property that for $q\supseteq p^{\prime}$, $q\in S(B)$, $Aa_{<N}\subseteq B$,
$F_{Mb}(q)\leq N$, so by the induction hypothesis we have that
$a_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{Aa_{<N}a_{N}\ldots a_{Nk}}a_{<i}$ for $i\geq
kN$.

∎

######  Remark 3.2.6.

When $p=\mathrm{tp}(a/A)$ where $a$ is a set of realizations of a regular type
$p_{0}\in S(B)$, $\mathrm{bdd}(C)=C\subseteq A$, some of these results hold
replacing $\mathrm{SU}$ with $\mathrm{SU}_{p_{0}}$ namely analogues of
Proposition 3.2.5 giving lower bounds on independence: when
$I=\\{a_{i}\\}_{i<\omega}$ is a Morley sequence in an extension $q$ of $p$ to
$B=\mathrm{bdd}(B)$ with $\mathrm{SU}_{p_{0}}(q)=n<\omega$,
$\mathrm{SU}_{p_{0}}(p)=n+1$, and
$\mathrm{SU}_{p_{0}}(\mathrm{Cb}(q)/A)>k(n+1)$ where $1\leq k<\omega$, then
$a_{k-1}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{<k-1}$.

Therefore, for

$N^{p_{0}}_{+}(p)=\mathrm{max}(\\{\mathrm{SU}_{p_{0}}(\mathrm{Cb}(q)/A):p\subseteq
q\in S(B),A\subseteq B=\mathrm{bdd}(B)\\})$

$N^{p_{0}}_{-}(p)=\mathrm{min}(\\{\mathrm{SU}_{p_{0}}(\mathrm{Cb}(q)/A):p\subseteq
q\in S(B),A\subseteq B=\mathrm{bdd}(B)\\})$

we have $N^{p_{0}}_{-}(p)\leq 2f_{Mb}(p)$, $N^{p_{0}}_{+}(p)\leq 2f_{Mb}(p)$
whenever $\mathrm{SU}_{p_{0}}(p)=2$.

Moreover, when $\mathrm{SU}_{p_{0}}(p)$ is finite and there is some $N$ so
that for all $q\supseteq p$, $q\in S(B)$, $A\subseteq B$, $F_{Mb}(q)\leq N$,
then there is some $N^{\prime}$ so that for all $q\supseteq p$, $q\in S(B)$,
$A\subseteq B$, $N^{p_{0}}_{+}(q)\leq N^{\prime}$. The reason why these proofs
work for $\mathrm{SU}_{p_{0}}$ is that, because $q$ and
$\mathrm{lim}^{+}(I/I)$ have a common nonforking extension, what we are really
showing is that when $F_{Mb}(q)\leq N$ , and
$\mathrm{SU}(p)-\mathrm{SU}(\mathrm{lim}^{+}(I/I))=1$ (or $\mathrm{SU}(p)=k$),
then $\mathrm{lim}^{+}(I/I)$ does not fork over $\\{a_{i}\\}_{i<N}$
($\\{a_{i}\\}_{i<kN}$), which implies the conclusion about the rank of
$\mathrm{Cb}(q)$, which satisfies
$\mathrm{bdd}(\mathrm{Cb}(q))=\mathrm{bdd}(\mathrm{Cb}(\mathrm{lim}^{+}(I)))$
. (The former case, where
$\mathrm{SU}(p)-\mathrm{SU}(\mathrm{lim}^{+}(I/I))=1$, is not Proposition
3.2.5 but is contained within the proof.) Since this will all happen inside of
$p_{0}$, the proof works for $\mathrm{SU}_{p_{0}}$ as well.

In rank $2$, one special case where the ranks of canonical bases are of
interest is that of plane curves, or minimal forking extensions of the type of
two independent realizations of a regular (or minimal) type. The following
extension of linearity was introduced for minimal types in [77], then
generalized to regular types in [86] (an excellent exposition of which can be
found in [12]; see [51] for further results on $k$-linearity); this is easily
seen to be equivalent to Definition 16 of [12].

######  Definition 3.2.7.

Let $p$ be a regular type over $A$. Then $p$ is $k$_-linear_ if, when
$p^{\prime}=\mathrm{tp}(ab/A)$ where $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{A}b$,
$a,b\models p$, every regular extension $q\supseteq p^{\prime}$, with $q\in
S(B)$, $A\subset B=\mathrm{bdd}(B)$, and $\mathrm{SU}_{p}(q)=1$, has
$\mathrm{SU}_{p}(\mathrm{cb}(q)/A)\leq k$, and moreover, there exists such a
type $p^{\prime}$ and extension $q$ with
$\mathrm{SU}_{p}(\mathrm{cb}(q)/A)=k$. The type $p$ is _linear_ if and only if
it is $1$-linear.

Note that in the case where $\mathrm{SU}(p)=1$, $\mathrm{SU}_{p}$ is the usual
$\mathrm{SU}$-rank and we can omit the assumption that $q$ is regular, because
$\mathrm{SU}(q)=1$ already implies that $q$ is regular.

######  Remark 3.2.8.

At least in the case of $\mathrm{SU}(p)=1$, $k$-linearity does not depend on
$p^{\prime}$ in the above definition when $p\in S(A)$ is a Lascar type: if
$p\otimes p$ is a type over $A$ realized by two independent realizations of
$p$ over $A$, and $p\otimes p$ has a forking extension $q$ with
$\mathrm{SU}(\mathrm{Cb}(q)/A)\geq k$, then for $(p\otimes p)^{\prime}$ some
other type over $A$ realized by two independent realizations of $p$ over $A$,
$(p\otimes p)^{\prime}$ has a forking extension $q^{\prime}$ with
$\mathrm{SU}(\mathrm{Cb}(q)/A)\geq k$. To see this, we apply the argument used
by [86] to get the group configuration from pseudolinearity; see Fact 3.2.11
below and the sketch of its proof. By Lascarness the independence theorem
holds, so there are $a,b,c$ independent over $A$ so that $ab\models p\otimes
p$, $bc\models p\otimes p$, $ac\models(p\otimes p)^{\prime}$. There are also
$c_{ab}$, $c_{bc}$ so that $\mathrm{SU}(c_{ab}/A)=k$,
$\mathrm{SU}(c_{bc}/A)=k$, $c_{ab}=\mathrm{Cb}(\mathrm{tp}(ab/c_{ab}))$,
$c_{bc}=\mathrm{Cb}(\mathrm{tp}(bc/c_{bc}))$; additionally we may choose
$c_{ab}$ and $c_{bc}$ so that $ac_{ab}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{Ab}cc_{bc}$. Let
$c_{ac}=\mathrm{cb}(ac/c_{ab}c_{ac})$; we show that $\mathrm{SU}(c_{ac}/A)\geq
k$, indeed that $\mathrm{SU}(c_{ac}/Ac_{ab})\geq k$. We see that
$c_{bc}\subseteq\mathrm{bdd}(c_{ab}c_{ac})$, since $b$ and $c$ are
interalgebraic over $c_{ab}c_{bc}$. But $c_{bc}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{A}c_{ab}$, so
$\mathrm{SU}(c_{bc}/Ac_{ab})=k$, so $\mathrm{SU}(c_{ac}/Ac_{ab})\geq k$.

We expect this is also true for $p$ a regular type, though we only really need
the above for Proposition 4.2, where $T$ is of finite rank and all regular
types have $\mathrm{SU}(p)=1$.

The following is then a consequence of Corollary 3.2.4, and the above Remark
3.2.8:

######  Corollary 3.2.9.

Let $p\in S(A)$ with $\mathrm{SU}(p)=1$. Then $p$ is $k$-linear if and only if
the maximum value of $F_{Mb}(p\otimes p)$ is equal to $k$, where $p\otimes p$
ranges over the types over $A$ realized by two independent realizations of
$p$. So if $p$ is a Lascar type (e.g. if $A=\mathrm{acl}^{eq}(A)$ and $T$ is
supersimple), and $p\otimes p$ is any type over $A$ realized by two
independent realizations of $p$ over $A$, $p$ is $k$-linear if and only if
$F_{Mb}(p\otimes p)=k$.

This just says that Theorem 3.3 (1) of [51], the equivalence of $k$-linearity
and being “$k$-based” for a Lascar strong type $p$, is witnessed by two
independent realizations of $p$; again, as stated above, we expect that its
proof makes use of similar ideas. In fact, using Theorem 3.3 (1) of [51], we
can do even better:

######  Fact 3.2.10.

Let $p\in S(A)$ with $\mathrm{SU}(p)=1$, and $n\geq 2$. Then $p$ is $k$-linear
if and only if the maximum value of $F_{Mb}(p^{(n)})$ is equal to $k$, where
$p^{n}$ ranges over the types over $A$ realized by $n$ independent
realizations of $p$. If $p$ is a Lascar type, and $p^{(n)}$ is any type over
$A$ realized by $n$ independent realizations of $p$ over $A$, then $p$ is
$k$-linear if and only if $F_{Mb}(p^{(n)})=k$.

###### Proof.

Direction (a) $\Rightarrow$ (b) of Theorem 3.3 (1) of [51] says that if $p$ is
$k$-linear and $I=\\{a_{i}\\}_{i<\omega}$ is an $A$-indiscernible sequence of
realizations of $p^{(n)}$, for $p^{(n)}$ any type realized by $n$ independent
realizations of $p$, then $I=\\{a_{k-1+i}\\}_{i<\omega}$ is a Morley sequence
over $Aa_{<k}$. (Note that Theorem 3.3 (1) of [51] says this explicitly only
when $p$ is a Lascar type, but the proof should extend to any complete type.)
Therefore, if $I$ is not a Morley sequence over $A$,
$a_{k}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{<k}$ and $F_{Mb}(p^{(n)})\leq k$. Since
$n\geq 2$, if $p$ is $k$-linear, there is some type $p^{(n)}$ realized by $n$
independent realizations of $k$ so that $F_{Mb}(p^{(n)})\geq k$; if $p$ is
additionally a Lascar type, this holds for any $p^{(2)}$ and thus any
$p^{(n)}$, by Remark 3.2.8.

∎

When $p$ is minimal, note that this definition is equivalent replacing
$\mathrm{SU}_{p}$ with just $\mathrm{SU}$.

In some classes of theories, $k=1$ is the only possible value for which a
regular type can be $k$-linearity. This is shown by [77], [86] and is highly
nontrivial, relying on the group configuration theorem in stable or
$\omega$-categorical simple theories.

######  Fact 3.2.11.

([77], [86]) Let $T$ be a stable theory and $p$ a regular type, or let $T$ be
an $\omega$-categorical simple theory, and let $p$ be a regular type over a
finite set. Then if $p$ is $k$-linear, $k=1$.

###### Proof.

(Sketch) We sketch the $\mathrm{SU}$-rank $1$ case. Suppose $p$ is $k$-linear.
Then as in Remark 3.2.8, we can get $a,b,c,c_{ab},c_{bc},c_{ac}$, where
$a,b,c$ have $\mathrm{SU}$-rank $1$ over $A$, and $c_{ab},c_{bc},c_{ac}$ have
$\mathrm{SU}$-rank $k$ over $A$; these will be arranged in a group
configuration, where any three of the $a,b,c,c_{ab},c_{bc},c_{ac}$ will be
independent, except for $a,b,c_{ab}$; $b,c,c_{bc}$; and $a,c,c_{ac}$. By [45]
in the stable case, [86] in the $\omega$-categorical case, we then obtain a
type-definable group of $\mathrm{SU}$-rank $k$ acting transitively on a type
of $\mathrm{SU}$-rank $1$. In the $\omega$-categorical case, [86] show that
there can be no such group when $k>1$; in the stable case, by Hrushovski’s
thesis, it is known that any group of $\mathrm{U}$-rank $k>1$ acting on a type
$p$ of $\mathrm{U}$-rank $1$ must be have $\mathrm{U}$-rank $1$, $2$ or $3$
(this proves a special case of the Borovik-Cherlin conjecture), and in the
case where the $\mathrm{U}$-rank is $2$ or $3$ the type $p$ is nonorthogonal
to a field, so can be shown not to be $k^{\prime}$-linear for any $k^{\prime}$
(see Example 3.1.2 for a related observation). ∎

From Fact 3.2.11, Corollary 3.2.9, and Remark 3.2.8 follows:

######  Corollary 3.2.12.

Let $p\in S(A)$ be a regular type in a stable theory, or a regular type over a
finite set in an $\omega$-categorical simple theory. Then either there is no
finite bound on $F_{Mb}(p\otimes p)$, where $p\otimes p$ ranges over all types
over $A$ realized by two independent realizations of $p$ over $A$, or $p$ is
linear.

Moreover, let $p\in S(A)$ be a regular _Lascar_ type, with $\mathrm{SU}(p)=1$
in a stable theory or an an $\omega$-categorical supersimple theory, and let
$p\otimes p$ be any type over $A$ realized by two independent realizations of
$p$ over $A$. Then either $p$ is not $k$-linear for any $k$ and
$F_{Mb}(p\otimes p)=\infty$, or $p$ is linear and $F_{Mb}(p\otimes p)=1$,

###### Proof.

The only case not yet clear from the facts quoted above is the case of a
Lascar type $p(x)\in S(A)$ in an $\omega$-categorical supersimple theory where
$A$ is not finite. (We include this case because it will be needed for
Proposition 4.2, showing that $F_{Mb}$ has values $1$ and $\infty$ when $T$ is
$\omega$-categorical supersimple of finite rank.) We may clearly replace
$A=\mathrm{bdd}(A)$. By supersimplicity, we may find some finite
$A^{\prime}_{0}\subset A$ over which $p(x)$ does not fork; we may also add the
($A_{0}$-definable, by $\omega$-categoricity) Lascar equivalence class of
realizations of $p(x)$ over $A^{\prime}_{0}$ to get a finite set
$A_{0}\subseteq A$; then $p(x)$ will not fork over $A_{0}$, and
$p(x)|_{A_{0}}$ will be a Lascar strong type over $A$ because all realizations
of $p(x)|_{A_{0}}$ will be Lascar equivalent over $A^{\prime}_{0}$, so over
$A_{0}$ because $A_{0}\subset\mathrm{acl}^{{eq}}(A^{\prime}_{0})$. So by the
independence theorem and the fact that $p(x)$ does not fork over $A_{0}$,
every type $p|_{A_{0}}(x)\otimes p|_{A_{0}}(x)$ realized by two
$A_{0}$-independent realizations of $p|_{A_{0}}(x)$ extends to a type
$p(x)\otimes p(x)$ realized by two $A$-independent realizations of $p(x)$, and
any type of the form $p(x)\otimes p(x)$ restricts to a type of the form
$p|_{A_{0}}(x)\otimes p|_{A_{0}}(x)$. So the case of $p(x)$ follows from the
case of $p(x)|_{A_{0}}$, a Lascar type over a finite set, by Corollary 3.2.9
and the below Claim 4.5. ∎

In Proposition 4.2 we will show that any type $p$ in a supersimple theory of
finite rank, whenever the conclusion of Fact 3.2.11 holds, must have
$F_{Mb}(p)=1$ or $F_{Mb}(p)=\infty$.

A good illustration of the case $F_{Mb}(p\otimes p)=\infty$ of Corollary
3.2.12 is given by $T=ACF_{0}$, the theory of algebraically closed fields of
characteristic zero. We saw in Example 3.1.2 that
$F_{Mb}\left(p\right)=\infty$, when $p\in S_{2}(\emptyset)$ is the generic
type of affine $2$-space.

######  Remark 3.2.13.

For an indiscernible sequence $I$ of realizations of $p\in S(A)$, the value
$\mathrm{SU}(\mathrm{Cb}(q)/A)$ when $T$ is supersimple (or of
$\mathrm{SU}_{p}(\mathrm{Cb}(q)/A)$ when $p$ is a regular type), where $q\in
S(B)$, $A\subseteq B=\mathrm{bdd}(B)$ is an extension of $p$ such that $I$ is
a Morley sequence over $B$ of realizations of $q$, can be easily seen to be an
invariant of $I$ that does not depend on $q$, without making reference to how
many terms of $I$ are independent over $A$.

When $T$ is simple but no longer supersimple, it will be of interest to
distinguish indiscernible sequences, but $\mathrm{SU}(\mathrm{Cb}(q)/A)$ may
no longer be defined as in the above remark. To circumvent this, we define
some invariants based on the “$\mathrm{G}$-rank,”
$\mathrm{SU}(\mathrm{Cb}(q)/A)$, of a canonical base. For our applications, we
will only need to define this for $\emptyset$-indiscernible sequences, though
the definitions can easily be generalized to any base.

######  Definition 3.2.14.

(1) Let $I$ be an indiscernible sequence over $\emptyset$. Then $I$ is
_$\kappa$ -resolvable_ if there is some set $A\subset\mathbb{M}$ (i.e. of real
elements), $|A|\leq\kappa$ so that $I$ is a Morley sequence over $A$.

(2) Let $p\in S(B)$ be a type. Then $p$ is _$\kappa$ -resolvable_ if there is
some set $A\subset\mathbb{M}$, $|A|\leq\kappa$ and $q\in S(AB)$, $q\supsetneq
p$ so that $q$ forks over neither $A$ nor $B$.

######  Definition 3.2.15.

Let $B\subset\mathbb{M}^{\mathrm{heq}}$. Then $G(B)$ (the “ _$\mathrm{G}$
-rank_” of $B$) is the least $\kappa$ so that there is $A\subseteq\mathbb{M}$,
$|A|\leq\kappa$ with $B\subseteq\mathrm{bdd}(A)$.

The following proposition, connecting these definitions, is an exercise:

######  Proposition 3.2.16.

Let $T$ be simple, and let $I$ be an indiscernible sequence. Then the
following are equivalent:

(1) The indiscernible sequence $I$ is $\kappa$-resolvable.

(2) For some type $p\in S(B)$, so that $I$ is a Morley sequence in $p$ over
$B$, $p$ is $\kappa$-resolvable.

(2’) For every type $p\in S(B)$, so that $I$ is a Morley sequence in $p$ over
$B$, $p$ is $\kappa$-resolvable.

(2”) The type $\mathrm{lim}^{+}(I/I)\in S(I)$ is $\kappa$-resolvable.

(3) For some type $p\in S(B)$, so that $I$ is a Morley sequence in $p$ over
$B$, $G(\mathrm{Cb}(p))\leq n$.

(3’) For every type $p\in S(B)$, so that $I$ is a Morley sequence in $p$ over
$B$, $G(\mathrm{Cb}(p))\leq n$.

(3”) $G(\mathrm{Cb}(\mathrm{lim}^{+}(I/I)))\leq n$

For our application to the Koponen conjecture in Section 6, we will need the
equivalence of (1), (2) and (2’); (3), (3’) and (3”) are included to justify
the analogy with the other invariants, and (3”) is included to obtain a
canonical invariant distinguishing indiscernible sequences.

In Section 2, we noted that for $p\in S(A)$ in a simple theory,
$\mathrm{nmdeg}(p)\leq F_{Mb}(p)$. The following notions, introduced in [30],
also generalize the degree of nonminimality:

######  Definition 3.2.17.

Let $p\in S(A)$ with $\mathrm{SU}(p)>\lambda$. Let the _$\lambda$ -forking
degree of $p$_, $F_{\lambda}(p)$, be the least $n$ so that there are
$a_{0},\ldots a_{n-1}\models p$, so that $p$ has a forking extension $q\in
S(Aa_{0}\ldots a_{n-1})$ with $\mathrm{SU}(q)\geq\lambda$.

It follows as in Remark 2.1.5 that $F_{\lambda}(p)$ is well-defined, and that

######  Proposition 3.2.18.

Let $p\in S(A)$ with $\mathrm{SU}(p)>\lambda$. Then $F_{\lambda}(p)\leq
F_{Mb}(p)$. So if $\mathrm{SU}(p)=\alpha$ is a limit ordinal and
$\mathrm{lim}_{\lambda\to\alpha}F_{\lambda}(p)=\infty$, $F_{Mb}(p)=\infty$.

######  Example 3.2.19.

The following example of a non-one-based theory, every type of which is
linear, appears in as Example 6.2.14 in [77]: let $T$ be the two-sorted theory
of an infinite-dimensional vector space $V$ over an algebraically closed field
$K$, in the language with the field operations, vector addition, and scalar
multiplication. This is superstable, with a single type $p\in
S^{1}_{V}(\emptyset)$ with $\mathrm{U}(p)=\omega$. The type $p$ is a linear
regular type, and the characterization of forking in [77] implies that for
$a_{0},\ldots a_{n}\models p$, $a_{n}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}a_{0}\ldots a_{n-1}$ if and only if $a_{0}$ is
in the linear span of $a_{0}\ldots a_{n-1}$; if $q\in S(a_{0}\ldots a_{n-1})$
is the generic of the linear span of $a_{0}\ldots a_{n-1}$, then
$\mathrm{U}(q)=n$. It follows that for $n<\omega$, $F_{n}(p)=n$, so in
particular, $\mathrm{lim}_{\lambda\to\omega}F_{\lambda}(p)=\infty$ and
$F_{Mb}(p)=\infty$.

An open problem, mentioned in [87], is whether there exists an
$\omega$-categorical supersimple theory of infinite $\mathrm{SU}$-rank. If $T$
is $\omega$-categorical and supersimple, let $p(x)\in S(A)$ have
$SU(p)=\alpha$, for $\alpha$ a limit ordinal; then
$\mathrm{lim}_{\lambda\to\alpha}F_{\lambda}(p)=\infty$. To see this, assume,
without loss of generality because $T$ is supersimple, that $A$ is finite, and
suppose $\mathrm{lim}_{\lambda\to\alpha}F_{\lambda}(p)=n$ for $n<\omega$; then
for $\lambda<\alpha$, there are $a_{0}\ldots a_{n}\models p$ with
$\mathrm{SU}(a_{n}/a_{0}\ldots a_{n-1}A)>\lambda$. Since $\lambda$ is a limit
ordinal, this implies there are infinitely many types in
$x_{0},\ldots,x_{n-1}$ over the finite set $S(A)$, contradicting
$\omega$-categoricity. So, we’ve shown:

######  Corollary 3.2.20.

Let $T$ be $\omega$-catgorical and supersimple, and $SU(p)$ a limit ordinal.
Then $F_{Mb}(p)=\infty$.

Note the similarity of this proof to that of Corollary 3.12 of Palacín [75],
exposited in Koponen [58].

##  4\. $F_{Mb}$ can be finite, but larger than $1$, in a stable theory

So far, we have only discussed examples of types $p\in S(A)$ where $p$ has the
property that has algebraic forking (Definition 2.1.7)–which fully
characterizes the forking extensions of $p$, and implies $F_{Mb}(p)=1$– or
where $F_{Mb}(p)=\infty$. This leads us to ask: in a simple theory, under what
conditions is it possible that $1<F_{Mb}(p)<\infty$, and when $F_{Mb}(p)=1$,
when must $p$ have algebraic forking?

We first show that in a supersimple theory with finite rank, the first
question is reducible to the following conjecture of Kim.

######  Conjecture 4.1.

(Pseudolinearity conjecture, [51].)

Let $T$ be simple, and let $p\in S(A)$ be a Lascar type with
$\mathrm{SU}(p)=1$. Then if $p$ is $k$-linear for some $k<\omega$, $p$ is
linear.

######  Theorem 4.2.

Let $T$ be a supersimple theory of finite rank. Then the following are
equivalent:

(1) $T$ satisfies the conclusion of the pseudolinearity conjecture.

(2) For every $p\in S(A)$, either $F_{Mb}(p)=1$ or $F_{Mb}(p)=\infty$.

Here, (2) $\Rightarrow$ (1) is just Corollary 3.2.9 and the definition of
$k$-linearity.

Note that in (2) of this proposition, we may assume $p$ is finitary777Recall a
hyperimaginary is finitary if it is the class of a finite tuple modulo a type-
definable equivalence relation.. We will see that 4.2 is essentially a
corollary of a powerful classical result of geometric stability theory, Fact
4.4. To state the form of the result we will need we first require a
definition:

######  Definition 4.3.

Two types $p,q\in S(A)$ are _equidominant_ over $A$ if there are realizations
$a$ and $b$ of $p$ and $q$ respectively if, for any set $B$,
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}B$ if and only if
$b\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}B$. Two complete types $p$, $q$ over any
sets are _equidominant_ if there is some set $A$ containing the domains of $p$
and $q$ so that there are nonforking extensions of $p$ and $q$ to $A$ which
are equidominant over $A$.

######  Fact 4.4.

Let $T$ be supersimple of finite rank, and $p\in S(A)$ a finitary type. Then
$p$ is equidominant with the type, over some set $B$, of a finite
$B$-independent set of realizations of minimal types over $B$.

The previous result follows from Corollary 5.2.19, Proposition 5.1.12, and
Lemma 5.2.11 of [87], the latter two of which imply that the regular types of
finite $\mathrm{SU}$-rank are exactly the types of $\mathrm{SU}$-rank $1$.

The theorem will then follow from the following three claims and Fact 4.4:

######  Claim 4.5.

Let $p$ and $q$ be complete types so that $q$ is a nonforking extension of
$p$. Then $F_{Mb}(p)=F_{Mb}(q)$.

######  Claim 4.6.

Let $p,q\in S(A)$ be equidominant over $A$. Then $F_{Mb}(p)=F_{Mb}(q)$.

######  Claim 4.7.

Let $p\in S(A)$ be the type of an $A$-independent set of realizations of
minimal types over $A$, and assume the conclusion of the pseudolinearity
conjecture holds. Than either $F_{Mb}(p)=1$ or $F_{Mb}(p)=\infty$.

We first prove Claim 4.5; this only requires simplicity. Let $F_{Mb}(p)=n$,
and choose an $A$-indiscernible sequence $I$, consisting of realizations of
$p$, any $n$ terms of which are independent over $A$, but which is not a
Morley sequence over $A$. By the chain condition in simple theories, and the
fact that $q$ does not fork over $p$, there is some $I^{\prime}\equiv_{A}I$,
with $I^{\prime}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}B$, consisting of realizations of $q$ and
indiscernible over $B$. But then any $n$ terms of $I^{\prime}$ are independent
over $B$, and $I^{\prime}$ is not a Morley sequence over $B$, so
$F_{Mb}(q)\geq n$. Now suppose $F_{Mb}(q)=n$; it suffices to show
$F_{Mb}(p)\geq n$. Let $I$ be an indiscernible sequence over $B$ of
realizations of $q$, that is not a Morley sequence over $B$, but so that any
$n$ terms of $I$ are independent over $B$. Since $q$ is a nonforking extension
of $p$, so each term of $I$ is independent from $B$ over $A$, it follows from
the forking calculus that every $n$ terms of $I$ (which consists of
realizations of $p$) are independent over $A$. But $I=\\{a_{i}\\}_{i<\omega}$
is not a Morley sequence over $A$, because if it were, for each $N<\omega$,
$\\{a_{Ni}\ldots a_{Ni+(N-1)}\\}_{i<\omega}$ would be a Morley sequence over
$A$ that remains indiscernible over $B$, so the first term of this would be
independent from $B$ over $A$ by Kim’s lemma and symmetry. So
$I\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}B$ (see, say, the proof of the independence
clause of Proposition 3.21 of [48]), and then since $I$ is assumed to be a
Morley sequence over $A$, it will be a Morley sequence over $B$, a
contradiction. Thus $F_{Mb}(p)\geq n$.

To prove Claim 4.6, supppose $F_{Mb}(p)=n$; by symmetry, it suffices to show
$F_{Mb}(q)\geq n$. Let $I=\\{a_{i}\\}_{i<\omega}$ be an indiscernible sequence
over $A$, $a_{i}\models p$, $a_{n-1}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{<n-1}$, $I$ not a Morley sequence. We
may then find $I^{\prime}=\\{a^{\prime}_{i}\\}_{i<\omega}$,
$a^{\prime}_{i}\models q$, so that for each $i<\omega$, $a_{i}$ forks over $A$
with exactly the same sets as $a^{\prime}_{i}$. Then it is easily seen by
successive replacements that $a^{\prime}_{n-1}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a^{\prime}_{<n-1}$, and $I^{\prime}$ is not
a Morley sequence. So $F_{Mb}(q)\geq n$.

Finally, it remains to prove Claim 4.7. There are two cases, first where all
of the independent minimal types are orthogonal, and second where some two are
not orthogonal. In the first case, this follows from Proposition 2.1.10 and
the following subclaim:

######  Subclaim 4.8.

Let $p(x),q(y)\in S(A)$ be orthogonal types and $r(x,y)$ the type over $A$ of
some (necessarily independent) realizations of $p$ and $q$. Then
$F_{Mb}(r)=\mathrm{max}(F_{Mb}(p),F_{Mb}(q))$.

###### Proof.

Obviously $F_{Mb}(r)\geq\mathrm{max}(F_{Mb}(p),F_{Mb}(q))$ (take a witness to
$F_{Mb}(p)$, and extend the terms independently to realizations of $r$; then
extract an $A$-indiscernible sequence.) To see
$F_{Mb}(r)\leq\mathrm{max}(F_{Mb}(p),F_{Mb}(q))$, let
$\\{a_{i}b_{i}\\}_{i<\omega}$ be an $A$-indiscernible sequence of realizations
of $r$ that is not a Morley sequence. It suffices to show that either
$I=\\{a_{i}\\}_{i<\omega}$ or $J=\\{b_{i}\\}_{i<\omega}$ is not a Morley
sequence over $A$, because, supposing without loss of generality that $I$ is
not a Morley sequence, $n+1$ terms of it will be dependent for $n\leq
F_{Mb}(p)$, so $n\leq\mathrm{max}(F_{Mb}(p),F_{Mb}(q))+1$ terms of
$\\{a_{i}b_{i}\\}_{i<\omega}$ will be dependent. If both $I$ and $J$ are
Morley sequences over $A$, then (see, say, [77]) $I\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{A}J$, so
$\\{a_{i}b_{i}\\}_{i<\omega}$ will be a Morley sequence over $A$, a
contradiction. ∎

This proves the case where the minimal types are orthogonal. When the minimal
types are nonorthogonal, we may assume, by the subclaim, that $p$ is realized
by an $A$-independent set $\\{a_{i}\\}_{i<n}$ for $n\geq 2$ where the
$p_{i}=\mathrm{tp}(a_{i}/A)$ all belong to the same nonorthogonality class; we
may assume $A=\mathrm{bdd}(A)$ (so we can apply the independence theorem). So
there is some set $B$ with $\mathrm{bdd}(B)=B$ and nonforking extensions
$q_{i}$ of $p_{i}$ to $B$ so that no two of the $q_{i}$ are weakly orthogonal,
and by the independence theorem we may additionally chose $B$ so that for each
$i$, $a_{i}\models q_{i}$, and $B\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}\\{a_{i}\\}_{i<n}$. By Claim 4.5, it
suffices to show that $F_{Mb}(\mathrm{tp}(\\{a_{i}\\}_{i<n}))$ is either $1$
or $\infty$. By Claim 4.6 we can replace each of the $a_{i}$ by tuples
interalgebraic over $B$, thereby assuming that each $a_{i}$ realizes the same
type over $B$. But then by the conclusion of the pseudolinerity conjecture
either this type is one-based, so $F_{Mb}(\mathrm{tp}(\\{a_{i}\\}_{i<n}))=1$,
or it is not $k$-pseudolinear for any $k$, so because $n\geq 2$, by Corollary
3.2.9, $F_{Mb}(\mathrm{tp}(\\{a_{i}\\}_{i<n}))=\infty$.

This completes the proof of Theorem 4.2. It follows from Corollary 3.2.12
that:

######  Corollary 4.9.

Let $T$ be either superstable of finite rank or supersimple of finite rank and
$\omega$-categorical. Then for $p\in S(A)$, either $F_{Mb}(p)=1$ or
$F_{Mb}(p)=\infty$.

Note that this proof does not tell us that types $p$ with $F_{Mb}(p)=1$ have
algebraic forking. For example, it is not known that algebraic forking is
preserved under nonforking extension, so the analogue of Claim 4.5 may not
hold. We may then ask:

######  Question 4.10.

Let $T$ be supersimple of finite rank and let $p\in S(A)$, $F_{Mb}(p)=1$. Does
$p$ have algebraic forking?

If the answer to this question is positive, by Propositions 2.1.10 and 4.2,
then, assuming the pseudolinearity conjecture, in a supersimple theory with
finite rank, the types $p$ with $F_{Mb}(p)$ finite will be exactly those with
algebraic forking.

######  Example 4.11.

There are superstable theories $T$ so that for all $p\in S(A)$, $F_{Mb}(p)=1$,
and which are not one-based, and therefore have types without algebraic
forking. In particular, there are superstable theories, which are not one-
based, which have the following property (See e.g. [77], where stable theories
with the below property are called “trivial”).

######  Definition 4.12.

A simple theory $T$ has _geometrically trivial forking_ if
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}bc$ implies $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}b$ or $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}c$. A regular type $p\in S(A)$ in a simple
theory has _trivial pregeometry_ if for $a,b,c\subseteq p(\mathbb{M})$,
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}bc$ implies $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}b$ or $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}c$.

The following is an obvious consequence of the definitions:

######  Proposition 4.13.

If $T$ is a simple theory with geometrically trivial forking, then
$F_{Mb}(p)=1$ for all complete types $p$. If $p$ is a regular type in a simple
theory with trivial pregeometry, then $F_{Mb}(p)=1$.

The $\omega$-stable free pseudoplane, Example 4.6.1 of [77], is an example of
a superstable theory with geometrically trivial forking which is not one-
based. This is the theory of undirected graphs without loops, each vertex of
which has infinite degree; it is the case $n=1$ of the construction of Theorem
4.19 below. As noted in example 7.2.10 of [77], the unique complete type $p$
over the empty set is regular, so has trivial pregeometry; however, the proof
from [77] that $T$ is not one-based should show that $p$ does not have
algebraic forking.

The Farey graph is another example of an $\omega$-stable theory (Corollary
1.15 of [23]) which is geometrically trivial (Lemma 1.14 of [23]) but is not
one-based (it has weak elimination of imaginaries, Corollary 11.15 of [23],
but $A\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}B$ does not coincide with
$\mathrm{acl}(AC)\cap\mathrm{acl}(BC)=\mathrm{acl}(C)$ for the real algebraic
closure, Lemma 11.4 of [23].)

However, even though there are examples of types $p$ in a supersimple theory
with $F_{Mb}(p)<\infty$ without algebraic forking, there are no known examples
of types $p$ in a supersimple theory with $F_{Mb}(p)$ other than $1$ or
$\infty$.

######  Question 4.14.

Let $T$ be supersimple, and $p$ a complete type. Is $F_{Mb}(p)$ either $1$ or
$\infty$?

We have also given two conditions under which a regular type $p$ in a
supersimple theory must have $F_{Mb}(p)=1$. Do these account for all examples?

######  Question 4.15.

Let $T$ be a supersimple theory, and $p$ a regular type with $F_{Mb}(p)=1$.
Must $T$ have either algebraic forking, or trivial pregeometry?

Though we do not have an answer to the question of whether $F_{Mb}(p)$ is $1$
or $\infty$ for $p$ a type in a supersimple (or even superstable) theory
(Question 4.14), we discuss this question further, in particular the case of
regular types. Regular types are an important special case in supersimple
theories, because of the fact (see e.g. Corollary 5.2.19 of [87], and compare
Fact 4.4) that every type in a supersimple theory is equidominant to the type
of a set of independent realizations of regular types. More precisely, if all
regular types $p$, and types $p$ of independent sets of realizations of a
linear regular type, have $F_{Mb}(p)=1$ or $F_{Mb}(p)=\infty$, we can use the
proof of Proposition 4.2 to prove that all types $p$ in superstable theories
have $F_{Mb}(p)=1$ or $F_{Mb}(p)=\infty$.

However, unlike types $p$ with $\mathrm{SU}(p)=1$, which have $F_{Mb}(p)=1$
because they have algebraic forking, it is not necessary that a regular type
$p$ has $F_{Mb}(p)=1$ (the unique type over the empty set in the vector space
sort in Example 3.2.19, for example.)

We give some characterizations of $F_{Mb}(p)$, where $p$ is a regular type. To
motivate this characterization, we reproduce the following definition from
[55], [1], which generalizes geometric triviality:

######  Definition 4.16.

Let $T$ be simple. Then $T$ has _$n$ -degenerate dependence_ if
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}B$ implies that, for some $B_{0}\subset B$
with $|B_{0}|\leq n$, $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}B_{0}$.

This suggests the following definition, where we assume that $B$ is a
sufficiently saturated model, rather than just any set. For the purposes of
our chracterization of $F_{Mb}(p)$ for $p$ regular, we present this relative
to a type in the theory $T$, fixing the base set $C$ as well as taking $a$ to
be a single realization of $p$.

######  Definition 4.17.

Let $T$ be simple, and $p(x)\in S(C)$. Then $p$ has _weakly $n$-degenerate
dependence over $C$_ if for $a\models p(\mathbb{M})$ and $|C|^{+}$-saturated
models $M\supset C$, if $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}p(M)$ then for some $B_{0}\subset p(M)$
with $|B_{0}|\leq n$, $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}B_{0}$.

######  Theorem 4.18.

Let $T$ be simple, and let $p\in S(C)$ be a regular type. Then the following
are equivalent:

(1) $F_{Mb}(p)\leq n$.

(2) $p$ has weakly $n$-degenerate dependence over $C$.

(3) For every forking extension $q$ over $p$, the solution set of $q$ is
covered by finitely many sets of the form $\mathrm{cl}_{p}(a_{0}\ldots
a_{n-1})$, where $a_{0},\ldots,a_{n-1}\models p$.

###### Proof.

(3 $\Rightarrow$ 2) Suppose (2) fails, so there is some $a\models p$ and
$|C|^{+}$-saturated model $M$ with $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}p(M)$, and $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{C}B_{0}$ for
every $B_{0}\subset p(M)$ with $|B_{0}|\leq n$. It suffices to show that
$q=\mathrm{tp}(a/p(M))$, which is a forking extension of $p$, fails the
conclusion of (2). Suppose otherwise, so there are $B^{0}_{0},\ldots
B_{0}^{k}\subset p(M)$ with $|B^{i}_{0}|\leq n$ for $0\leq i\leq k$, so that
$q(\mathbb{M})\subseteq\cup^{k}_{i=0}\mathrm{cl}_{p}(B^{i}_{0})$. Then $a\in
q(\mathbb{M})$, so $a\in\mathrm{cl}_{p}(B^{i}_{0})$ for some $0\leq i\leq k$,
so $a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}B^{i}_{0}$, a contradiction.

(1 $\Rightarrow$ 3) Suppose the negation of (3) holds, for some forking
extension $q\in S(C^{\prime})$ of $p$. We first construct inductively a
sequence $\\{a_{i}\\}_{i<\omega}$, $a_{i}\models q$, so that any $n+1$ terms
of $\\{a_{i}\\}_{i<\omega}$ are independent over $C$. Suppose
$\\{a_{i}\\}_{i<k}$ is a sequence, though of length $k$, satisfying these same
requirements. We want to find $a_{k}\models q$ that is independent from every
$n$-element subset of $\\{a_{i}\\}_{i<k}$, thereby preserving the
requirements. Let $B^{0}_{0},\ldots B_{0}^{m}\subset p(M)$ enumerate the
$n$-element subsets of $\\{a_{i}\\}_{i<k}$. Then by assumption,
$q(\mathbb{M})\not\subseteq\cup^{m}_{i=0}\mathrm{cl}_{p}(B^{i}_{0})$; choosing
$a_{k}\in q(\mathbb{M})\backslash\mathrm{cl}_{p}(B^{i}_{0})$ will give us the
desired $a_{k}$.

Now choose a $C^{\prime}$-indiscernible sequence $\\{a^{\prime}_{i}\\}_{i<k}$
with the same EM-type over $C$ as $\\{a_{i}\\}_{i<k}$. The condition that any
$n+1$ terms are independent over $A$ is type-definable (see Subclaims 7.12 and
7.16), so $\\{a^{\prime}_{i}\\}_{i<k}$ also has this condition. However,
$\\{a^{\prime}_{i}\\}_{i<k}$ is not a Morley sequence, since if
$\\{a^{\prime}_{i}\\}_{i<\omega}$ were a Morley sequence over $C$, by Kim’s
lemma and symmetry $a^{\prime}_{0}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}C^{\prime}$, contradicting the fact that
$a^{\prime}_{0}\models q$, a forking extension of $p$. So since
$\\{a^{\prime}_{i}\\}_{i<k}$ is an $n+1$-independent sequence of realizations
of $p$ that is not a Morley sequence, $F_{Mb}(p)>n$.

(1 $\Rightarrow$ 2). Suppose that $p$ has $n$-degenerate independence over
$C$, and that $F_{Mb}(p)>n$. Then there is a $C$-indiscernible sequence
$\\{a_{i}\\}_{i<\omega}$, $a_{i}\models p$, so that for $N>n$, $a_{0},\ldots
a_{N-1}$ form an independent set over $C$ but $a_{i}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}a_{0}\ldots a_{N-1}$ for $i>N-1$. The sequence
$\\{b_{i}\\}_{i<\omega}$ where $b_{i}=a_{i+N-1}$ has, by $C$-indiscernibility
of $\\{a_{i}\\}_{i<\omega}$, the property that $b_{0},\ldots b_{N-1}$ forms an
independent set over $C$, and is indiscernible over $C^{\prime}=Ca_{0}\ldots
a_{N-1}$, so we may choose some $|C|^{+}$-saturated model $M\supset
C^{\prime}$ so that $\\{b_{i}\\}_{i<\omega}$ remains indiscernible over
$Cp(M)$. By $n$-degenerate independence and indiscernibility over $p(M)$,
there is then some $B_{0}\subset p(M)$ with $B_{0}\leq n$ so that
$b_{i}\in\mathrm{cl}_{p}(B_{0})$, in particular for $0\leq i\leq N-1$. But
because $SU_{p}(B_{0})\leq n$, this contradicts independence of $b_{0},\ldots
b_{N-1}$.

∎

The direction (3 $\Rightarrow$ 1) of the above suggests a new way of obtaining
$n$-independent indiscernible sequences that are not Morley sequences,
possibly giving a strategy for answering Question 4.14; note that its proof
uses a different construction method than the arguments in the results on
canonical bases in the previous section (Proposition 3.2.3, Corollary 3.2.4),
where a Morley sequence is taken in a forking extension of $p$.

We now give our first examples of types $p$ with $1<F_{Mb}(p)<\infty$. In
general, it seems quite difficult to construct such examples - for instance,
since our constructions in Section 5 are either superstable or
$\omega$-categorical supersimple of finite rank, by Corollary 4.9 every type