$p$ in these theories has either $F_{Mb}(p)=1$ or $F_{Mb}(p)=\infty$. We show
that for any $n<\omega$, it is possible in a stable theory that $F_{Mb}(p)=n$.

######  Theorem 4.19.

Let $n<\omega+1$. Then there is an stable theory $T$ with a type $p$ so that
$F_{Mb}(p)=n$, and so that, for all types $q$, $F_{Mb}(q)\leq n$.

Fix $n<\omega$. We define $T$, based on the free projective planes discussed
by Hyttinen and Paolini ([46]); for $n=1$ this will coincide with the free
$\omega$-stable pseudoplane (see e.g. [77]). In fact, this theory will itself
work for $n=2$, and we expect that the open generalized $n$-gons of [2] will
work for any value of $n$. However, these examples are more complicated than
necessary; for example, in the free projective planes of [46], the algebraic
closure is determined not just by the requirement that the incidence relation
has no finite subgraphs of minimum degree at least $n+1$, but also the
requirement that there is exactly one line incident to any two points, and
vice versa. We only need a requirement of the first kind to obtain the desired
theory.

Call an undirected graph open if it does not contain a finite subgraph of
minimum degree at least $n+1$. For $A\subseteq B$ open graphs and $C\subseteq
B$ finite, call $C$ closed over $A$ if every point of $C\backslash A$ has at
least $n+1$ neighbors in $C$; write $A\leq B$ if there is no $C\subseteq B$,
$C$ not contained in $A$, which is closed over $A$. (Here we borrow the
terminology of [46].) If $C\leq A$, $C\leq B$ and $A$ and $B$ are freely
amalgamated over $C$, $C\leq AB$: under these assumptions, if $D\subseteq AB$
is closed over $C$, then so is $D\cap A$ and $D\cap B$, and if $D$ contains
points not in $C$ then so does one of these two. By the same reasoning, $AB$
is an open graph.

Let $T$ be the theory of infinite open undirected graphs, every $n$ nodes of
which have infinitely many common neighbors. Then, just as the free projective
planes of Hall are models of the theory of open projective planes discussed in
[46], $T$ is consistent: at stage $0$, start with an infinite independent set,
and at stage $k+1$, for each $n$ nodes of stage $k$, add infinitely many nodes
connected to only those nodes. The following generalizes straightforwardly the
content of the proof of Theorem 1.1 of [46]:

######  Fact 4.20.

Let $M\models T$ be saturated, and let $A,B$ be small open undirected graphs
so that $A\leq M$, $A\leq B$. Then there is an embedding $\iota:B\to M$, which
is the identity on $A$, so that $\iota(B)\leq M$.

Let us work in a sufficiently saturated model $\mathbb{M}\models T$. It
follows from this fact, and the discussion immediately preceding the
definition of $T$, that if $A\leq\mathbb{M}$, $A$ is algebraically closed, and
if $A^{\prime}\leq\mathbb{M}$ is isomorphic to $A$, $A\equiv A^{\prime}$. In
fact, a set $A\subseteq\mathbb{M}$ is algebraically closed if and only if
$A\leq M$; to see this, it suffices to show that if $A$ is an open undirected
graph, and $A\subsetneq B$ with $B\backslash A$ a finite set closed over $A$ ,
then there cannot be an open graph $C\supset A$ containing infinitely many
distinct $B_{i}$ isomorphic to $B$ over $A$. But by Ramsey’s theorem, we may
assume that there is some fixed $D\supseteq A$ so that $B_{i}\cap B_{j}=D$ for
$i\neq j$ and all of the $B_{i}$ are isomorphic over $D$. Let $D_{0}$ be the
(finite) set of neighbors of some point of $B_{i}\backslash D$ for some
(every) $i$. Then we see that for distinct $i_{1}\ldots i_{n+1}$,
$F=\cup^{n+1}_{j=1}(B_{i}\backslash D)\cup D_{0}$ is a finite set so that each
point of $f$ has at least $n+1$ neighbors in $F$, contradicting the fact that
$C$ is an open graph.

Now let $A,B,C\subseteq\mathbb{M}$ be sets, where $C\leq\mathbb{M}$. We show
that $A\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}B$ (where $\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 5.71527pt\scriptscriptstyle{}}}$ denotes
forking independence) if and only if
$\mathrm{acl}(ABC)=\mathrm{acl}(AC)\cup\mathrm{acl}(BC)$ and
$\mathrm{acl}(AC)\cup\mathrm{acl}(BC)$ is the free amalgamation of
$\mathrm{acl}(AC)$ and $\mathrm{acl}(BC)$ over $C$; denote the latter
condition $A\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}^{\otimes}B$. Since forking independence is
then stationary over models, $T$ is stable. This is similar to the
characterization of forking in free projective planes in [46], but we include
it to illustrate a property of forking that may not be familiar from other
examples of stable theories.

First, if $A\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{\otimes}_{C}B$ with $C\leq M$, then
$A\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}B$ because clearly, $\mathrm{tp}(A/CB)$
extends to a $B$-invariant global type. Now suppose that
$A\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}B$. Then, $A\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}\mathrm{acl}(BC)$, so we may assume $C\leq
B$ and $B$ is algebraically closed. Now, let $A^{\prime}=\mathrm{acl}(AB)$ and
suppose that $A\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}^{\otimes}B$. We show that
$A\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}B$. We may produce a $C$-indiscernible
sequence $\\{B_{i}\\}_{i<\omega}$ with $B_{0}=B$ so that, for $i<\omega$,
$B_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}^{\otimes}\\{B_{j}\\}_{j<i}$; for
$p(X,B)=\mathrm{tp}(A/B)$, it suffices to show that
$\cup\\{p(X,B_{i})\\}_{i<\omega}$ is inconsistent. Suppose otherwise; then we
may in fact assume that $\\{B_{i}\\}_{i<\omega}$ is indiscernible over
$A^{\prime}$. We then get a contradiction from the following claim

######  Claim 4.21.

(Here we still assume $A\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}^{\otimes}B$ with $C\leq B$, $B$
algebraically closed.) Let $\\{B_{i}\\}_{i<\omega}$ be a
$A^{\prime}=\mathrm{acl}(AC)$-indiscernible sequence with $B_{0}=B$. Then
$B_{n}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}^{\otimes}\\{B_{i}\\}_{i<n}$.

###### Proof.

We may assume that $B_{i}\cap B_{j}=C$ for $i\neq j<\omega$; then clearly,
$A^{\prime}\cap B=C$. It will suffice to show that
$\mathrm{acl}(\\{B_{i}\\}_{i<n+1})\supsetneq\cup_{i<n+1}B_{i}$.

So 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}^{\otimes}B$, either
$\mathrm{acl}(A^{\prime}B)=A^{\prime}B$ and $A^{\prime}$ and $B$ are not
freely amalgamated over $C$, or $\mathrm{acl}(A^{\prime}B)\supsetneq
A^{\prime}B$. In the first case, there is some $a\in A^{\prime}\backslash C$
with a neighbor $b$ in $B\backslash C$. Then $\\{a\\}$ will have $n+1$
neighbors in $\\{B_{i}\\}_{i<n+1}$, so will belong to
$\mathrm{acl}(\\{B_{i}\\}_{i<n+1})$ but will not belong to
$\cup_{i<n+1}B_{i}$. In the second case, since $\\{B_{i}\\}_{i<\omega}$ and
thus $\\{\mathrm{acl}(A^{\prime}B_{i})\\}_{i<\omega}$ is indiscernible over
$A^{\prime}$, there is some fixed $D\supseteq A^{\prime}$ so that
$\mathrm{acl}(A^{\prime}B_{i})\cap\mathrm{acl}(A^{\prime}B_{j})=D$ for $i\neq
j\leq\omega$, and so that $\\{B_{i}\\}_{i<\omega}$ is indiscernible over $D$.
As we are in the second case, there are also $\\{F_{i}\\}_{i<\omega}$, so that
$F_{i}\subseteq\mathrm{acl}(A^{\prime}B_{i})\backslash A^{\prime}B_{i}$ is
finite and closed over $A^{\prime}B_{i}$ for $i<\omega$, and so that
$\mathrm{acl}(A^{\prime}B_{i})F_{i}\equiv\mathrm{acl}(A^{\prime}B_{j})F_{j}$
for $i\neq j<\omega$. Suppose first that for some (every) $i<\omega$,
$F_{i}\subseteq D$. Each of the $F_{i}$ must have neighbors in $B_{i}$, since
it is closed over $A^{\prime}B_{i}$ and $A^{\prime}$ is algebraically closed.
So if $F_{i}\subseteq D$, then $D$ and $B$ are not freely amalgamated over
$C$, and $B_{n}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}^{\otimes}\\{B_{i}\\}_{i<n}$ by the first
case. Otherwise, for some (every) $i<\omega$, $F_{i}\not\subseteq D$. There is
some finite $D_{0}\subseteq D$ so that, for all $i<\omega$, $D_{0}$ consists
of all neighbors of $F_{i}\backslash D$ in $D$. Then
$F=D_{0}\cup^{n}_{i=0}(F_{i}\backslash D)$ is a finite set, not contained in
$\cup_{i<n+1}B_{i}$, so that every point of $F$ has at least $n+1$ neighbors
in $F\cup(\cup_{i<n+1}B_{i})$: each point of $F_{i}$ has $n+1$ neighbors that
are either in $D\supseteq A$ (so in $D_{0}\subseteq F$), in $F_{i}\backslash
D$, or in $B_{i}$, while each point of $D_{0}$ has a neighbor in each of the
$F_{i}\backslash D\subseteq F$, and therefore has at least $n+1$ neighbors in
$F$. So $\mathrm{acl}(\\{B_{i}\\}_{i<n+1})\supsetneq\cup_{i<n+1}B_{i}$, as
$F\subseteq\mathrm{acl}(\\{B_{i}\\}_{i<n+1})$.

∎

We have proven that $T$ is stable, 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}^{\otimes}B$ characterizing forking over
algebraically closed sets. We can now show that for $p\in S(A)$ a nonalgebraic
type, $F_{Mb}(p)\leq n$: we may assume that $A$ is algebraically closed and
that $p=\mathrm{tp}(B/A)$ for $A\leq B$ and $B$ algebraically closed. Let
$\\{B_{i}\\}_{i<\omega}$ be an $A$-indiscernible sequence, with $B_{i}\models
p$ for $i<\omega$, and suppose it is not a Morley sequence over $A$. We must
show that $B_{n}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}\\{B_{i}\\}_{i<n}$. But there is some $N$
so that $B_{j}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}^{\otimes}\\{B_{i}\\}_{i<N}$ for all $j\geq
N$, because $\\{B_{i}\\}_{i<\omega}$ is not a Morley sequence over $A$. Since
$\\{B_{i+N}\\}_{i<\omega}$ is indiscernible over
$\mathrm{acl}(\\{B_{i}\\}_{i<N})$, by the claim,
$B_{n+N}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}^{\otimes}\\{B_{i+N}\\}_{i<n}$, so
$B_{n}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}\\{B_{i}\\}_{i<n}$, as desired. Finally, we
find a type $p\in S(\emptyset)$ with $F_{Mb}(p)\geq n$, so $F_{Mb}(p)=n$. Let
$A=\\{a_{i}\\}\cup\\{*\\}$ be an undirected graph with the $a_{i}$, $*$
distinct, where there is an edge between each of the $a_{i}$ and $*$ but no
edges between the $a_{i}$. Then there is an embedding $\iota:A\to\mathbb{M}$
with algebraically closed image; for $i<\omega$, let $b_{i}=\iota(a_{i})$.
Then $\\{b_{i}\\}_{i<\omega}$ is an $\emptyset$-indiscernible sequence, so
that $b_{n-1}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}\\{b_{i}\\}_{i<n-1}$ but
$b_{n}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}\\{b_{i}\\}_{i<n}$. Therefore, $F_{Mb}(p)\geq
n$, as desired, proving Theorem 4.19.

##  5\. $\operatorname{nmdeg}$ and $F_{ind}$ in categorical theories

In this section, we give examples showing that the degree of nonminimality of
types can be any $n\in\mathbb{N}$ for $\omega$-categorical theories that are
supersimple or strictly stable, and for almost strongly minimal theories. This
solves the analog of a problem of the second author and Moosa in the context
of simple theories. Examples of Freitag and Moosa ([35, discussion following
Theorem C, page 5], [33] and [32]) previously showed that the degree of
nonminimality can be as large as two in a stable theory.

Before giving the constructions, we explain the results of [33] and its
implications for constructing such examples. In [33], it is shown that for
finite rank types in a theory with the property that _any non-locally modular
minimal type is non-orthogonal to a minimal type over the emptyset_ 888Such as
the theory of differentially closed fields or compact complex manifolds.,
$\operatorname{nmdeg}(p)\geq d$ implies that the type is internal to a non-
locally modular minimal type and has a binding group which is generically
$d$-transitive [32, Section 2]. In the case that the theory satisfies the
Zilber trichotomy, this binding group and its action on the type is isomorphic
to the action of an algebraic group acting regularly on an algebraic variety;
the arguments of [33] can then be used to show that the degree of
nonminimality is at most two. Even without the trichotomy assumption, to a
great extent, by the O’Nan-Scott theorem in this setting [64], the binding
group actions can be (to a large extent) reduced to actions of simple groups
on the left cosets of definable subgroups (see e.g. Theorem 2.3 of [34]).

So, any finite rank types with $\operatorname{nmdeg}(p)\geq n>2$ involve
either:

  1. (1)

A theory in which there are nonorthogonality classes of minimal types $p$ such
that for any type in the class, $F_{cb}\left(p\right)>2$.

  2. (2)

A definable simple group serving as a binding group which has transitivity
characteristics not possessed by algebraic group actions.999Specifically,
analyzing the arguments of [33], a high degree of generic transitivity and the
property that for large values of $n$, every non-generic orbit of the binding
group on $p^{n}$ is finite.

A (finite rank) $\omega$-stable example along the lines of 2 would involve a
counterexample to the Cherlin-Zilber conjecture, so we pursue a example along
the lines of 1 via a Hrushovski construction of non-locally modular regular
types. Indeed the most natural method to get large degree of nonminimality
along the lines of 1 involves ensuring that we have minimal types $p$ such
that for any minimal type $q$ in the nonorthogonality class of $p$,
$F_{cb}\left(q\right)=n$ for building a type with degree of nonminimality $n$.

Lemma 5.3.25 shows there are simple theories of $r$-spaces where
$k-1$-transitivity implies there exist types $p$ with
$\operatorname{nmdeg}(p)$ arbitrarily large (depending on the theory). We use
quite different variants of the Hrushovski construction to build such examples
that are $\aleph_{1}$ but not $\aleph_{0}$-categorical,
$\aleph_{0}$-categorical simple, or strictly stable $\aleph_{0}$-categorical).

###  5.1. Hrushovski constructions

We outline the properties of two constructions Hrushovski gave in the late
1980’s to refute a) Zilber’s trichotomy conjecture for strongly minimal sets
and b) Lachlan’s conjecture that an $\aleph_{0}$-categorical stable theory is
$\aleph_{0}$-stable. The key to both examples is to define a ‘pre-dimension’
that describes a family of matroids. These examples will all be
$r$-hypergraphs. The unique relation symbol $R$ in the vocabulary holds only
of distinct $r$-tuples and in any order.

The constructions are all from amalgamation classes of finite structures.

######  Definition 5.1.1 ($\prec$-amalgamation classes).

A $\prec$-amalgamation class $({\bf K}_{0},\prec)$ is a collection of finite
structures for a vocabulary $\sigma$ (which may have function and relation
symbols) satisfying:

  1. (1)

$\prec$ is a partial order refining $\subseteq$.

  2. (2)

$\prec$ satisfies joint embedding and amalgamation.

  3. (3)

$A,B,C\in{\bf K}_{0}$, $A\prec B$, and $C\subseteq B$ then $A\cap C\prec C$.

  4. (4)

${\bf K}_{0}$ is countable

Condition Definition 5.1.1.3 is not essential for the next result. However it
follows when the amalgamation class arises from a dimension function as in
Definition 5.1.3 and is used for the later analysis.

######  Theorem 5.1.2.

For a $\prec$-amalgamation class, there is a countable structure $M$, the
$\prec$-generic model, which is a union of members of ${\bf K}_{0}$, each
member of ${\bf K}_{0}$ embeds in $M$, and $M$ is $\prec$-homogeneous.

For Fraïssé, the language is finite relational, the class is closed under
substructure, and $\prec$ is $\subseteq$. We describe below two instances of
$\prec$ used in this paper.

The notion of pre-dimension and dimension establish a matroid structure on the
generic model.

######  Definition 5.1.3.

Work in a relational vocabulary $R$. Let $\delta:{\bf
K}_{0}\rightarrow\mathcal{N}$ be a function with $\delta(\emptyset)=0$ that
give the properties of a matroid. Extend $\delta$ to $d:{\hat{\bf
K}}\times{{\bf K}}_{0}\rightarrow\Re^{+}$ by for each $N\in{\hat{\bf K}}$,
$d_{N}(A)=\inf\\{\delta(B):A\subseteq B\subseteq_{\omega}N\\}$.

Require for $A,B,C\in{\bf K}_{0}$ that are subsets of $N\in\hat{\bf K}$.

  1. (1)

$d_{N}(A/C)\geq 0$.

  2. (2)

$d_{N}(AB/C)=d_{N}(A/BC)+d_{N}(B/C)$.

  3. (3)

If $A\subseteq A^{\prime}$ then $d_{N}(A^{\prime}/C)\geq d_{N}(A/C)$
(equivalent to submodularity).

  4. (4)

For every $A$ and there is a finite $A_{0}$ with $A\subseteq A_{0}\subseteq N$
such that $\delta(A_{0})=d_{N}(A)$.

The original examples were of the form $\epsilon(A)=m|A|-n|R(A)|$, where
$|R(A)$ is the number of instances of tuples satisfying $\tau$ relations in
$A$. A necessary extension for the counterexample to Lachlan’s conjecture is
to allow a carefully chosen irrational coefficient.

Strong Substructure

Each conjecture requires a different interpretation of $\prec$ on ${\bf
K}_{0}$.

######  Definition 5.1.4.

  1. (1)

For Zilber conjecture: $A\leq B$ (strong substructure) if $\forall B^{\prime}$
with $A\subset B^{\prime}\subset B$, $\delta(B^{\prime}/B^{\prime}\cap A)\geq
0$.

  2. (2)

For Lachlan conjecture: $A_{\leq_{*}}B$ (*-strong substructure) if $\forall
B^{\prime}$ with $A\subset B^{\prime}\subset B$,
$\delta(A)<\delta(B^{\prime})$:

  3. (3)

$\mathop{\rm icl}(A)$ is the minimal extension $B$ of $A$ with $B\leq N$ for
any $N\in\hat{\bf K}$. It can be obtained as union of a chain $B_{i}$ such
that $\delta(B_{i+1}/B_{i})<0$, is contained in $\operatorname{acl}(A)$ and is
$\leq$ any extension in ${\bf K}_{0}$. Without the control functions discussed
below $|\mathop{\rm icl}(A)|$ may not be bounded in terms of $|A|$ [9, 8, 62].

[27, p 159] calls the first interpretation $\leq$ and the second $\leq^{*}$;
[26] calls the first $\leq_{s}$ and the second $\leq_{d}$. We follow the
earlier notation of [9].

######  Definition 5.1.5.

$M\in\hat{\bf K}_{0}$ has finite closures with respect to $({\bf
K}_{0},\prec)$ if for every finite $A\subseteq M$ there is a finite $B\prec M$
with $A\subseteq B$.

For rational coefficients both $\leq$ and $\leq_{*}$ have finite extensions
that are ‘strong – i.e. ‘closed in the universe’, but for $\leq$ such
extensions are not bounded in size while it unique for $\leq_{*}$. For
irrational coefficients $\leq$ need not have finite closures.

The crucial distinction between the two is that while for rational
coefficients both satisfy Definition 5.1.3.iv), only $\leq_{*}$ has a unique
minimal $*$-strong extension which can then be thought of as a
$*$-closure101010This is often, somewhat misleadingly, called a $d$-closure,
since it is the choice of strong, not of $d$ which determines the precise
closure..

The position of the theory of the generic in the stability classification
depends on rational vs irrational coefficients of $\delta$ and on the choice
between $\leq$ and $\leq_{*}$. Here $\mathop{\rm icl}(A)$ is the minimal
extension $B$ of $A$ with $B\leq N$ for any $N\in\hat{\bf K}$. We will write
generic for $\leq$-generic and $*$-generic for $\leq_{*}$-generic.

######  Definition 5.1.6 (d-independence).

$c$ is forking independent from $B$ over $A\subseteq B$ if

  1. (1)

[9, §3] For $\leq$: $d(c/B)=d(c/A)$ and $\mathop{\rm icl}{cA}\cap\mathop{\rm
icl}{B}\subseteq\mathop{\rm icl}{A}$.

  2. (2)

[27, Cor. 2.20] $\leq_{*}$: $d(c/B)=d(c/A)$ and $\operatorname{acl}(cA)\cap
B=A$.

The minimal $B\supseteq A$ with $B\leq M$ ($\mathop{\rm icl}(B)$ can be
strictly smaller than the minimal $B\supseteq A$ with $B\leq_{*}M$. The second
$B=\operatorname{acl}(A)$ in the $\aleph_{0}$-categorical (rational
coefficients) $\leq_{*}$-case.

When the choice of $\prec$ is clear we write $c\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{C}^{d}B$ in
either case.

######  Lemma 5.1.7.

${\bf R}(A)$ of instances of the relation of $R$ in $A$ (more generally the
sum over finitely many relations).

  1. (1)

$\leq$ [9, 3.39]: The theory of an ab initio generic for $d(A)=\alpha|A|-{\bf
R}(A)$ is stable if $\alpha$ is irrational and $\omega$-stable if rational.

  2. (2)

$\leq$ [27, 2.20,2.24,3.9]: The theory of a *-generic model of a
$\aleph_{0}$-categorical theory for $d(A)=\alpha|A|-{\bf R}(A)$ is stable if
$\alpha$ is irrational and supersimple of finite rank if rational.

Lemma 5.1.7 1) and 2) agree for irrational $\alpha$ since then
$\leq=\leq_{*}$.

Evans works with expansions of an underlying $\aleph_{0}$-categorical theory
but we restrict to the ab initio case, beginning with a collection of finite
structures.

###  5.2. Supersimple and strictly stable $\aleph_{0}$-categorical theories

In this section we study various Hrushovski constructions on the class ${\bf
K}_{0}$ of $k$-spaces to obtain $\aleph_{0}$-categorical theories of various
stability classes with maximal $\operatorname{nmdeg}(p)=k$. The distinction
will arise from different choices of the pre-dimension $d_{0}$, control
function, and notion of strong substructure.

In [29], Evans and Pantano construct $\omega$-categorical simple theories
which they show are $n$-transitive. In an $n$-transitive simple theory, if
$a_{1},\ldots a_{n-1}\models p$ for $p\in S(\emptyset)$ a type with
$\mathrm{SU}(p)>1$, then there must be exactly one nonalgebraic type over
$a_{1}\ldots a_{n-1}$, because the types of any $n$ distinct realizations of
$p$ are all the same. So this nonalgebraic $q\in S(a_{1}\ldots a_{n-1})$ must
be a nonforking extension of $p$, so $\mathrm{nmdeg}(p)\geq n$. We expect that
the construction of [29] has finite $\mathrm{SU}$-rank and $\mathrm{SU}$-rank
greater than $1$, giving us an example of an $\omega$-categorical supersimple
theory of finite rank with a type $p$ with $\mathrm{nmdeg}(p)>n$. However,
their construction is more complicated than necessary to get $n$-transitivity
(because they also have the goal of making the algebraic closure grow
arbitrarily fast), so we show that we can also get $\mathrm{nmdeg}(p)\geq n$
in the less complicated constructions of [27].

In [68], there is an overview of the constructions of Evans, which we
reproduce here in modified form. Let $\mathcal{L}$ be a language with finitely
many relations, and for each relation symbol $R_{i}$, let $\alpha_{i}$ be such
that $\alpha\cdot n_{i}$ is a non-negative integer associated to $R_{i}$,
where $n_{i}$ is the arity of $R_{i}$. For $A$ a finite
$\mathcal{L}$-structure, define a predimension
$d_{0}(A)=|A|-\sum_{i}\alpha_{i}|R_{i}(A)|$, with $R_{i}(A)$ the set of tuples
of $R_{i}$ with elements of $A$. As usual in Hrushovski constructions of
hypergraphs, we write $|R(A)|$ for the number of sets realizing the relation
$R$. As in Definition 5.1.4, define the relation $A\leq_{*}B$ for $B$ any
$\mathcal{L}$ -structure and $A$ a finite substructure of $B$, to mean that
every finite superstructure of $A$ within $B$ has predimension greater than
$A$. Let $f$ be an increasing continuous positive real-valued function and let
$\mathcal{C}_{f}$ be the class of finite $\mathcal{L}$-structures
$A^{\prime}$, so that the $R_{i}$ are irreflexive (i.e., only contain tuples
of distinct elements) and symmetric (invariant under permutation of
coordinates) on $A^{\prime}$, and any substructure $A\subseteq A^{\prime}$
satisfies $d_{0}(A)\geq f(A)$. (Note that symmetry is not required by [27],
but is useful for the presentation here.)

######  Fact 5.2.1.

For any $n<\omega$, there is an $\omega$-categorical supersimple theory
$T_{n}$ admitting a type $p$ with $\mathrm{nmdeg}(p)\geq n$.

###### Proof.

Let $T_{n}$ be the theory of the *-generic $M$ for the class $\mathcal{C}_{f}$
where $f(x)=\frac{\mathrm{log}(x)}{\mathrm{log}(3)}$ with $\tau=\\{R\\}$, a
$k$-ary relation, with $k=3^{2n+1}$.

We show that $T$, constructed as in [27, §3] as in the exposition of [27],
suffices, when $\mathcal{L}=\\{R\\}$ for $R$ a $k$-ary relation with $k=n+1$,
$\delta(A)=2|A|-\frac{|R(A)|}{k}$, and $f$ is defined as follows: $f(x)=2x-2$
for $x\leq n+2$,
$f(x)=2x-2+(\frac{\mathrm{log}(x)}{\mathrm{log}(3)}-\frac{\mathrm{log}(n+2)}{\mathrm{log}(3)})$
for $x\geq n+2$

Now the criteria for $T$ being well-defined and simple,
$0<f^{\prime}(x)<\frac{1}{x}$ and $f(3x)\leq f(x)+1$, will hold for all $x\geq
n+2$, but not for all $x\geq 0$. However, as in Remark 3.8 of [27], we may
individually check the instances of amalgamation where some of the structures
being amalgamated have size at most $n+1$, since these structures either have
no relations, or have size $n+1$ and one relation. Therefore, $T$ is a well-
defined, $\omega$-categorical simple theory. Moreover, because $f(x)\leq 2x$
for all $x>0$, and $2x-1\geq f(x)\geq 2n$ for $x\geq k$, we again see that
there is only one type in $S_{n-1}(\emptyset)$ realized by an $n-1$-tuple of
distinct elements. In particular, there is a unique type $p\in
S_{1}(\emptyset)$. As before, we see that $\mathrm{SU}(p)=2$, and that
$\mathrm{nmdeg}(p)\leq k-1$, so in this case $\mathrm{nmdeg}(p)\leq k-1=n$.

To see $\mathrm{nmdeg}(p)\geq n$, so $\mathrm{nmdeg}(p)=n$, we show there is a
unique type in $S_{n}(\emptyset)$ realized by distinct $a_{1},\ldots,a_{n}$.
The given lower bound on $f$ implies $f(x)>2n$ for $x>n+2$, so
$|\mathrm{acl}(a_{1},\ldots,a_{n})|\leq n+1$. But $|\mathrm{acl}(a_{1}\ldots
a_{n})|=n+1$ is impossible, because by choice of $n$, if $|A|=n+1=k$,
$\delta(A)\geq 2n+1>\delta(a_{1}\ldots a_{n})$.,

Now to show that $n=\mathrm{nmdeg}(p)<F_{ind}(p)$, note that if $\langle
a_{i}<\omega\rangle$ is a nonconstant indiscernible sequence either every
$n+1$-tuple satisfies $R$ or none does. The first is clearly impossible since
long enough such sequences have negative predimension. In the second case,
$d_{0}(a_{1}\ldots a_{n+1})=2n+2$. Because $f(x)>2n+2$ for $x>n+2$,
$|\mathrm{acl}(a_{1}\ldots a_{n+1})|\leq n+2$, so either $a_{1},\ldots
a_{n+1}$ is closed or $|\mathrm{acl}(a_{1}\ldots a_{n+1})|=n+2$ and
$d_{0}(\mathrm{acl}(a_{1}\ldots a_{n+1}))\geq f(n+2)=2n+2$. For either option
$a_{n+1}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}a_{1},\ldots a_{n}$ and $F_{ind}(p)>n$. ∎

We modify this example to show the strict inequality
$n=\mathrm{nmdeg}(p)<F_{ind}(p)$ for arbitrary $n$.

######  Theorem 5.2.2.

Let $n<\omega$. There is an $\omega$-categorical supersimple theory $S_{n}$
admitting a type $p$ with $\mathrm{nmdeg}(p)=n$ and
$F_{ind}(p)>\mathrm{nmdeg}(p)$.

###### Proof.

Let $S_{n}$ be the theory of the *-generic $M$ for a class ${\bf K}_{f}$ where
$k=n+1$ (i.e. $R$ is $n+1$-ary) and $f(x)=2x-2$ for $x\leq n+2$,
$f(x)=2x-2+(\frac{\mathrm{log}(x)}{\mathrm{log}(3)}-\frac{\mathrm{log}(n+2)}{\mathrm{log}(3)})$
for $x\geq n+2$.

Now the criteria for free amalgamation, $0<f^{\prime}(x)<\frac{1}{x}$ and
$f(3x)\leq f(x)+1$, will hold for all $x\geq n+2$, but not for all $x\geq 0$.
However, for the same choice of $d_{0}$ as in Theorem 5.2.1, free amalgamation
still holds. Since, as in Remark 3.8 of [27], we may individually check the
instances of amalgamation where some of the structures being amalgamated have
size at most $n+1$, since these structures either have no relations, or have
size $n+1$ and one relation. Moreover, as $f(x)\leq 2x$ for all $x>0$, and
$2x-1\geq f(x)\geq 2n$ for $x\geq k$, we again see that there is only one type
in $S_{n-1}(\emptyset)$ realized by an $n$-tuple of distinct elements, and
that the unique type $p\in S_{1}(\emptyset)$ has $\mathrm{SU}(p)=2$, whence
$\mathrm{nmdeg}(p)\leq k-1$, so here $\mathrm{nmdeg}(p)\leq k-1=n$.

To see $\mathrm{nmdeg}(p)\geq n$, so $\mathrm{nmdeg}(p)=n$, we show there is a
unique type in $S_{n}(\emptyset)$ realized by distinct $a_{1},\ldots,a_{n}$.
The given lower bound on $f$ implies $f(x)>2n$ for $x>2$, so
$|\mathrm{acl}(a_{1},\ldots,a_{n})|\leq n+1$. But as in Theorem 5.2.1
$|\mathrm{acl}(a_{1}\ldots a_{n})|=n+1$ is impossible.

Now to show that $n=\mathrm{nmdeg}(p)<F_{ind}(p)$, note that if $\langle
a_{i}<\omega\rangle$ is a nonconstant indiscernible sequence either every
$n+1$-tuple satisfies $R$ or none does. The first is clearly impossible since
long enough such sequences have negative predimension. In the second case,
$d_{0}(a_{1}\ldots a_{n+1})=2n+2$. Because $f(x)>2n+2$ for $x>n+2$,
$|\mathrm{acl}(a_{1}\ldots a_{n+1})|\leq n+2$, so either $a_{1},\ldots
a_{n+1}$ is closed or $|\mathrm{acl}(a_{1}\ldots a_{n+1})|=n+2$ and
$d_{0}(\mathrm{acl}(a_{1}\ldots a_{n+1}))\geq f(n+2)=2n+2$. For either option
$a_{n+1}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}a_{1},\ldots a_{n}$ and $F_{ind}(p)>n$. ∎

We apply Evan’s ([27]) modification of the construction used by [41], [42] to
obtain the original non-superstable examples of $\omega$-categorical stable
theories. He avoids the requirement that $\alpha$ has infinite index. Evans’s
more general criteria for $T$ to be a well-defined simple theory are as
follows: define $\kappa(x)$, for $x>0$, to be the minimum of $1$ and the
smallest possible positive value of $\delta(D)-\delta(A)$ where $A\leq_{*}D$
and $|D|\leq x$ (Definition 3.1 of [27].) Using the more general statement of
Lemma 3.3 and Theorem 3.6(ii) of [27], $T$ is well-defined and simple whenever
$0<f^{\prime}(x)<\frac{\kappa(x)}{x}$ and $f(3x)\leq f(x)+\kappa(x)$. Again,
we may apply Remark 3.8 of [27]: we will be able to show that $T$ is a well-
defined simple theory when $f(x)=2x-2$ for $x\leq n+2$, and on $x\geq n+2$,
$f$ is any continuous function satisfying the criteria just listed. Then the
proof that $T$ has a type $p$ with $\mathrm{nmdeg}(p)=n$ and
$F_{ind}(p)>\mathrm{nmdeg}(p)$ is as above; the only difference is that the
$\mathrm{SU}$-rank of $T$ will no longer be $2$, because it is no longer the
case that $d(a/\emptyset)$ can only strictly decrease twice when enlarging the
base set. However, in the case where $\alpha$ is irrational, one sees that the
following condition holds: $A\supsetneq B$, then $\delta(A)\neq\delta(B)$. By
Lemma 3.9 of [27], whenever $\delta$ satisfies this condition, $T$ (if well-
defined) is stable, regardless of $f$. So the proof of Theorem 5.2.2 also
gives us:

######  Theorem 5.2.3.

For every $n<\omega$ there is an $\omega$-categorical strictly stable theory,
admitting a type $p$ with $\mathrm{nmdeg}(p)=n$ and
$F_{ind}(p)>\mathrm{nmdeg}(p)$.

The theory is not superstable as an $\aleph_{0}$-categorical superstable
theory is $\omega$-stable and then must be locally modular. But the Hrushovski
examples are CM-trivial.

###  5.3. $\aleph_{1}$-categorical theories

######  Theorem 5.3.1.

For each $r,k<\omega$ there is an $\aleph_{1}$-categorical (almost strongly
minimal) theory $T_{r,k}$ that has an ( $1$-type $p\in S(\emptyset)$ with
$\operatorname{nmdeg}(p)=r$; the nonforking extension of $p$ over $r$ elements
is strongly minimal. Each model of cardinality $\kappa$ is an
$r-(\kappa,\kappa,1)$ design and the restriction to a strongly minimal subset
$P$, is a $r-(\kappa,k,1)$ design ($k$-Steiner system).

######  Definition 5.3.2 (Curves and Designs).

A $\tau$-structure $(M,R)$ is

  1. (1)

An $r$-space if $R$ defines an $r$-hypergraph in which each $r-1$ -tuple of
distinct points in contained in unique maximal $R$-clique (curve).

(Equivalently, $A$ is a $t-(v,\lambda,1)$ design is a set $Q$ of points, of
cardinality v, together with a set $\mathcal{B}$ of blocks (curves) each of
which has cardinality $\lambda$, and which has the property that any set of
$t$ points is a subset of only $1$ block.

This is enforced by requiring that for any $r$-space $A$, if $a$ and $b$ are
each on a line through $\bf c=\langle c_{0},c_{1},\ldots c_{r-1}\rangle$ then
$R(a,b,\bf c^{\prime})$ for any $\bf c^{\prime}\subset\bf c$ with $\bf
c^{\prime}=r-2$.

  2. (2)

${\bf K}^{*}_{0}={\bf K}^{*}_{0}(r)$ denotes the collection of all finite
$r$-spaces.

  3. (3)

An $r$-space is a $(r,k)$-Steiner system if all curves have the same length
$k$.

For infinite $\lambda$ we will build $(r-1)-(\lambda,\lambda,\lambda)$ block
designs in the $\aleph_{0}$-categorical case and both
$(r-1)-(\lambda,\lambda,\lambda)$ and $(r-1)-(\lambda,k,\lambda)$ (for fixed
$k$) in the $\aleph_{1}$-categorical case.

######  Definition 5.3.3.

(Degree of nonminimality) Suppose $p\in S(A)$ is a nonalgebraic and nonminimal
stationary type. By the degree of nonminimality of p, denoted by
$\operatorname{nmdeg}(p)$, we mean the least k such that p has a nonalgebraic
forking extension over $A\cup\\{a_{1},\ldots,a_{k}\\}$, for some
$a_{1},\ldots,a_{k}$ realising p.

So for $m<\operatorname{nmdeg}(p)$, if $\mathop{\rm lg}({\bf a})=m$ is a
sequence of realizations of $p$ and $q$ with $p\subseteq q\in S(A{\bf a})$
forks over $A$, $q$ has only finitely many realizations.

By a $t-(v,k,\lambda)$ design we mean a set $Q$ of points, of cardinality v,
together with a set $\mathcal{B}$ of blocks each of which is a subset with
cardinality $k$ of $Q$, and which has the property that any set of t points is
a subset of exactly $\lambda$ blocks.

The Hrushovski construction [43] created examples of strongly minimal setd
whose geometries were neither discrete, group-like nor field-like. The
modifications in [5, 7, 6] to obtain $\aleph_{1}$-categorical theories of non-
Desarguesian geometries/Steiner systems, and Morley rank 2 fields with
definable infinite subsets, respectively are adapted here.

A _H_ rushovski sm-class depends on the choice of a quintuple $(\sigma,{\bf
K}_{0},\epsilon,{\bf L}_{0},{\bf U})$ of parameters. The role of the
parameters are expounded in [10, §1.1]; here $\sigma,{\bf L},\epsilon,{\bf U}$
are replaced by $\tau,{\bf K},\delta,\mathcal{U}$ chosen for the current
problem. Each parameter is different from any earlier argument.

Now we define our geometrically based pre-dimension function by modifying
[76]. Although the following definitions each depend on $r$, since we will fix
$r$ for each example, after describing the vocabulary, we have suppressed the
$r$. The unary predicate $P$ is used to guarantee that the theory is almost
strongly minimal and so $\aleph_{1}$-categorical. This allows a $k$-Steiner
system on $P$ while the lines are infinite in the entire structure.

######  Definition 5.3.4.

We define the appropriate $\tau,{\bf K}^{*}$ and ${\bf K}_{0}$.

  1. (1)

$\tau$ has an $r$-ary relation $R$ and a unary predicate $P$.

  2. (2)

Every $(A,R,P)\in{\bf K}^{*}_{0}$ is a finite $r$-space.

  3. (3)

For $(A,R)\in{\bf K}^{*}$ let:

$\delta(A)=2(|A|)-|P(A)|-\sum_{\ell\subseteq A}\mathbf{n}_{A}(\ell).$

  4. (4)

$(A,R)\in{\bf K}^{\prime}_{0}$ if for any $A^{\prime}\subseteq
A,\delta(A^{\prime})\geq 0$.

  5. (5)

$(A,R)\in{\bf K}_{0}$ if $A\in{\bf K}^{\prime}_{0}$ and there is no
$B\subseteq A$ containing a subset $C$ of a single realization of $R$ with
$\delta(B/C)<\delta(C)$.

The switch from ${\bf K}^{\prime}_{0}$ to ${\bf K}_{0}$ in Definition 5.3.4 is
patterned on [43, Proposition18]; ${\bf K}_{0}$ has amalgamation just as ${\bf
K}^{\prime}_{0}$ does, as the amalgamation procedure introduces none of the
forbidden structures.

https://sites.pitt.edu/~kaveh/Chap7-MATH1050.pdf (page 6) gives an example of
a $3-(8,4,1)$ design, $A$. Note that there is a copy of $A\subseteq\neg P$ in
${\bf K}^{\prime}_{0}$ as for such an $A$, no subset of $A$ has negative rank,
and $\delta(A)=2$ while the $4$ element curve has rank $7$ and so is not
strong in $A$. Thus, the restriction to ${\bf K}_{0}$ is essential for the
transitivity. Another design on page 8 of the same chapter is $3-(10,4,1)$
with 30 blocks; thus it is is forbidden in ${\bf K}^{\prime}_{0}$, even in
$\neg P$.

######  Definition 5.3.5.

  1. (1)

We denote the cardinality of an $R$-clique $\ell\subseteq A\in{\bf K}^{*}_{0}$
by $|\ell|$, and, for $B\subseteq A$, we denote by $|\ell|_{B}$ the
cardinality of $\ell\cap B$.

  2. (2)

We say that a non-trivial curve $\ell$ contained in $A$ is based in
$B\subseteq A$ if $|\ell\cap B|\geq d$, in this case we write $\ell\in L(B)$.

  3. (3)

The nullity of a line $\ell$ contained in a structure $A\in\mathbf{K}^{*}_{0}$
is:

$\mathbf{n}_{A}(\ell)=|\ell|-(r-1).$

Note that since $R$ is symmetric, we can picture a model as a plane with a
subset $X$ such that for each $a\in X$ and each $b_{1},\ldots b_{r-1}$,
$R(a,b_{1},\ldots b_{r-1})$

######  Definition 5.3.6 (Strong Extensions).

  1. (1)

For any class ${\bf K}$ of finite structures, $\hat{\bf K}$ denotes the
collection of structures of arbitrary cardinality that are direct limits of
models in ${\bf K}$.

  2. (2)

Extend $\delta$ to $d:{\hat{\bf K}}_{0}\times{{\bf K}_{0}}\rightarrow\omega$
by for each $N\in{\hat{\bf K}}_{0}$ and $A\subset_{<\omega}N$,
$d(N,A)=\inf\\{\delta(B):A\subseteq B\subseteq_{\omega}N\\}$,
$d_{N}(A/B)=d_{M}(A\cup B)-d_{M}(B)$. We usually write $d(N,A)$ as $d_{N}(A)$
and omit the subscript $N$ when clear.

$d_{N}(A)=d_{N}(A/\emptyset).$

  3. (3)

For any $N\in\hat{\bf K}_{0}$ and finite $B\subseteq N$, we write $B\leq N$
(read $N$ is a _strong extension_ of $B$) when $B\subseteq A\subseteq N$
implies $d_{N}(A)\geq d_{N}(B)$.

  4. (4)

We write $B<A$ to mean that $B\leq A$ and $B$ is a proper subset of $A$.

  5. (5)

A structure $A$ is an $({\bf K}_{0},\leq)$-generic or $\leq$-homogeneous if
$A$ is a $({\bf K}_{0},\leq)$-union and for any $B\leq C$ each in ${\bf
K}_{0}$ and $B\leq A$ there is a $\leq$-embedding of $C$ into $A$.

In this case, if $A\subseteq M$ and $|A|\leq r-1$ then for any $B\supseteq A$,
$A\leq B$. In other situations, we can restrict the finite structures by
adding such a condition.

######  Observation 5.3.7.

Counting automorphism types: If $|X|=s\leq r-1$, $X$ is strong in any
extension. Thus, in the generic model the quantifier free type of $X$
determines the automorphism type of $X$; a priori there are $2^{s}$ such types
depending which elements of $X$ are in $P$. If $|X|=r$, the other quantifier
free question asks whether $R$ or $\neg R$ holds of the tuple, yielding
$2^{r+1}$ $r$-types. Note however that since the truth of $R({\bf a})$ does
not depend on the order of ${\bf a}$, there are actually only $r+1$
automorphism types of $r$-tuples depending on the number of $a_{i}$ in $P$.

######  Definition 5.3.8.

Canonical Amalgamation For any class $({\bf K}_{0},\delta)$, if $A\cap B=C$,
$C\leq A$ and $A,B,C\in{\bf K}^{*}$, $G$ is a free (or canonical)
amalgamation, $G=B\oplus_{C}A$, if $G\in{\bf K}^{*}$,
$\delta(A/BC)=\delta(A/C)$ and $\delta(B/AC)=\delta(B/C)$

It follows [7, Lemma 3.10] that the amalgam satisfies:
$\delta(A\oplus_{C}B)=\delta(A)+\delta(B)-\delta(C)$ and any $D$ with
$C\subseteq D\subseteq A\oplus_{C}B$ is also free. Thus, $B\leq G$. In
addition to submodularity, [7] proves that the constructed strongly minimal
set is ‘flat’ ([43]).

The crux of $\aleph_{1}$-categoricity is the ‘algebrizing’ function $\mu$
which makes all non-generic types in $P$ algebraic.

######  Definition 5.3.9.

Let $A,B\in{{\bf K}}_{0}$ with $B\subseteq A$ and111111For ease of exposition,
we insist primitive extensions are proper. $B-A\neq\emptyset$.

  1. (1)

$A$ is a _primitive extension_ of $B$ if $B\leq A$ and there is no
$B\subsetneq A_{0}\subsetneq A$ such that $B\leq A_{0}\leq A$. $A$ is a
$k$-primitive extension if, in addition, $\delta(A/B)=k$.

  2. (2)

We say that the $0$-primitive pair $A/B$ is good if there is no
$B^{\prime}\subsetneq B$ such that $(A/B^{\prime})$ is $0$-primitive. ([43]
called this a minimal simply algebraic or m.s.a. extension.)

  3. (3)

If $A$ is $0$-primitive over $B$ and $B^{\prime}\subseteq B$ is such that we
have that $A/B^{\prime}$ is good, then we say that $B^{\prime}$ is a base for
$A$ (or sometimes for $A-B$).

  4. (4)

If the pair $A/B$ is good, then we also write $(B,A)$ is a good pair.

######  Definition 5.3.10.

[The algebrizing function]

  1. (1)

Let $\mathcal{U}$ be the collection of functions $\mu$ assigning to every
isomorphism type ${\beta}$ of a good pair $C/B$ in ${\bf K}_{0}$ a natural
number $\mu({\beta})=\mu(B,C)\geq\delta(B)$.

  2. (2)

For any good pair $(B,C)$ with $B\subseteq M$ and $M\in\hat{\bf K}_{0}$,
$\chi_{M}(B,C)$ denotes the number of disjoint copies of $C$ over $B$ in $M$.
A priori, $\chi_{M}(B,C)$ may be $0$.

  3. (3)

Let ${\bf K}_{\mu}$ be the class of structures $M$ in ${\bf K}_{0}$ such that
if $(B,C)$ is a good pair then $\chi_{M}(B,C)\leq\mu((B,C))$

Minor modifications of the proof in [7] yield.

######  Theorem 5.3.11.

Let $\mu\in\mathcal{U}$; the class ${\bf K}_{\mu}$ has amalgamation and thus a
generic model $M$.

Since each set of size $\leq r$ is strong in the generic model,

######  Lemma 5.3.12.

The isomorphism type of an $s$-element set ${\bf a}$ for $s\leq r$ is
determined by the number of elements of $P$ among the $a_{i}$.

We now describe an additional parameter $k$ for $\mu$ which guarantees that
each line in $M$ intersects $P(M)$ in $(r+k)$-points.

######  Notation 5.3.13.

Let $\beta_{s}$ be the isomorphism type of $s\leq r$ elements and let
$\beta^{-}_{s}$ denote the isomorphism type of the structure obtained by
dropping any single point.

Since $R$ is symmetric it is harmless to regard the dropped point as the last
one.

######  Observation 5.3.14.

  1. (1)

If $s<r$, $\delta(\beta/\beta^{-})>0$

  2. (2)

If $s=r$ and the last element is not in $P$, $\delta(\beta/\beta^{-})>0$.

  3. (3)

If $s=r$ and the last element is in $P$, then $\delta(\beta/\beta^{-})=0$ and
$\beta/\beta^{-}$ is a good pair.

######  Definition 5.3.15.

Let ${\bf U}_{t}$ be the subset of $\mathcal{U}$ such that if $\beta$ is one
of the $2^{r-1}$ isomorphism types of a sequence satisfying Observation
5.3.14.3, then $\mu(\beta/\beta^{-})=t$.

######  Observation 5.3.16.

If $\mu\in{\bf U}_{t}$ then each line in the generic model $M$ for ${\bf
K}_{\mu}$ has $k=r+t-1$ elements in $P$.

While lines in models of $T_{\mu}$ are infinite, each line intersects $P(M)$
in $k$ points, for $k$ as in Notation 5.3.16, motivating the following
notation.

######  Notation 5.3.17.

If we fix the arity of the relation $R$ as $r$ and $\mu$ is in $U_{t}$ with
$k=r+t$, We write $T_{r,k,\mu}$ for the theory of the generic model for ${\bf
K}_{\mu}$.

######  Observation 5.3.18.

The restriction of $T_{r,k,\mu}$ to $P$ is the analog (with larger $r$ and
line length) to the strongly minimal Steiner system in a vocabulary with a
single ternary relation of [7]. So $T\upharpoonright P$ is strongly minimal.

$T_{r,k,\mu}$ is the theory of generic for ${\bf K}_{\mu}$ with this choice of
$\mu$.

######  Definition 5.3.19.

A $t-(v,k,\lambda)$ design is a set $P$ of $v$ points, together with a set
$\mathcal{B}$ of blocks each of which is a $k$-subset of $P$, and which has
the property that any set of $t$ points is a subset of exactly $\lambda$
blocks.

The design description just summarises the $r$-space property and the results
on curve length.

######  Lemma 5.3.20.

Any two distinct curves intersect in $r-2$ points.

###### Proof.

In the class ${\bf K}_{0}$, given $A$, a set of $2r-4$ distinct points on two
non-intersecting lines; if $B=Ab$ where $a$ is on each line then $B$ is in
${\bf K}_{0}$ since $\delta(Ba)>\delta(B)$. Thus in the generic (by ‘what can
happen does’) any two lines intersect. But the intersection has at most $r-2$
points since the lines are distinct and $r-1$ points determine a line.∎

Recall that a theory $T$ is almost strongly minimal [11] if there is a finite
set of constants ${\bf a}$ realizing a principal type and a strongly minimal
formula $\phi$ such that for every $N\models T$, $N=\operatorname{acl}({\bf
a}\cup\phi(M)$.

######  Theorem 5.3.21.

The theory $T_{r,k,\mu}$ is almost strongly minimal and for $M\models
T_{r,k,\mu}$ with $|M|=\kappa$. $P(M)$ is a strongly minimal $r-1$-Steiner
system (an $(r-1)-(\kappa,k,1)$ design), and $(M,R)$ is an
$(r-1)-(\kappa,\kappa,1)$-design.

###### Proof.

Let $M$ be the countable generic model and fix two equal length sequences
${\bf a},\bf b$ that make up $2r-4$ distinct points not in $P(M)$ (from the
prime model). Note that for each $c\in M$, the lines ${\bf a}c$ and $\bf bc$
intersect $P$ in $\\{c^{1},c^{2}\\}$ (where each $c^{i}$ is one of the
$k$-points in $P(M)$ on ${\bf a}c_{i}$ and $c$ is one of the $r-1$ points on
the intersection of ${\bf a}c^{1}$ and $\bf bc^{2}$. (Without loss, $c^{1}\neq
c^{2}$ since the number of points in $P$ on either line is $k>r-2$.) Thus,
$c\in\operatorname{acl}({\bf a},\bf b,c^{1},c^{2})$ and $T$ is almost strongly
minimal.

∎

There are in fact finitely many codes for $c$ (e.g. $|\\{{\bf a},c\cap
P\\}|=k+r-a$), but this doesn’t matter for interalgebraicity. More precisely,

######  Definition 5.3.22.

An $(n,m)$-relation $T$ on $X\times Y$ is definable binary relation such that
each $x$ is related to $m$ elements of $Y$ and each $y$ is related to $n$
elements of $X$.

######  Lemma 5.3.23.

Each curve defined by a formula $R({\bf a},x)$ ($|{\bf a}|=r-1$) is strongly
minimal.

###### Proof.

Fix $r-2$ points $\bf c$ in $\neg P$. Now, as $b$ ranges through $R({\bf
a},x)$, there must be a point $d\in P(M)$ ($d\neq b$ since $k>1$) with $d,b$
both on the intersection of the curves determined ${\bf a}$ and $\bf c$. By
Lemma 5.3.20 the two curves intersect in $r-3$-further points and any of the
$k$ points on $R(\bf c,y,b)\wedge P(y)$ determines (with $\bf c$ the curve
through $\bf cd$. Thus, the formula $\phi(x,y)=(R({\bf a},x)\wedge R(\bf
c,y,x)\wedge P(y)$ defines a $(k,r-2)$ relation between and $P(y)$ and $R({\bf
a},x)$; so $R({\bf a},x)$ is also strongly minimal. ∎

We can choose $\bf c\in P(M)$, guaranteeing that the generic types of $P(M)$
and $R(\bf c,M)$ are not weakly orthogonal.

######  Theorem 5.3.24.

For $T_{r,k,\mu}$ the complete type over $\emptyset$, $p$, given by $\neg
P(x)$ has $\operatorname{nmdeg}(p)=r-1$ and
$F_{ind}(p)>\operatorname{nmdeg}(p)$.

###### Proof.

Choose distinct $a_{1},a_{2},a_{3},\ldots a_{r}$ realizing $p(v)$ and such
that $R(a_{1},a_{2},a_{3},\ldots a_{r})$. Let
$p_{i}=\operatorname{tp}(a_{i}/A_{<i})$. Each $p_{i}$ for $i<r$ does not fork
over the empty set because $\mathop{aut}(M)$ acts $i$-transitively on $\neg P$
guaranteeing each $p_{i}$ is the unique non-forking extension of $p$ to its
domain $A_{<i}$; The type $p_{r}$ forks over $\emptyset$ because there are
infinitely many distinct choices of $a_{r-1}^{j}$ giving different lines. Any
pair of the lines $\ell_{i}=\langle a_{1},a_{2}\dots a_{r-1},a^{j}_{r}\rangle$
intersect only in $a_{1},a_{2}\dots a_{r-1}$ since any further point of
intersection would imply the lines are identical. But then the formula

For $F_{ind}$, let $\langle a_{i}:i<\omega$ be indiscernible sequence over the
empty set realizing $p$. Then either every $r$-tuple realizes $R$ or none
does. In the first case, we have an indiscernible sequence such that
$\operatorname{tp}(a_{n}/a_{<n}$ witnesses
$F_{ind}(p)>\operatorname{nmdeg}(p)$. ∎

Note that by Corollary 4.9, the $p$ in Theorem 5.3.24 satisfies
$F_{Mb}(p)=\infty$. But by Corollary 3.2.2 $F_{ind}(p)\leq 2(r-1)$.

For r-spaces we have a general argument from r-transitivity to
$\operatorname{nmdeg}=(r-1)$.

######  Lemma 5.3.25.

Suppose $T$ is a simple theory of a $r$-space, $p\in S(\emptyset)$ with
$\mathrm{SU}(p)=n<\omega$, and for any model $M$ of $T$, $\mathop{aut}(M)$
acts $r-1$ transitively on $p(M)$. Then $\operatorname{nmdeg}(p)=r-1$.

###### Proof.

Choose distinct $a_{1},a_{2},a_{3},\ldots a_{r}=A_{<r+1}$ realizing $p$ and
such that $R(a_{1},a_{2},a_{3},\ldots a_{r})$. Let
$p_{i}=\operatorname{tp}(a_{i}/A_{<i})$. Each $p_{i}$ for $i<r$ does not fork
over the empty set because $\mathop{aut}(M)$ acts $i$-transitively on $p(M)$
guaranteeing each $p_{i}$ is the unique non-forking extension of $p$ to its
domain; the only forking extensions over $A_{<i}$ are realized in $A_{<i}$.
The type $p_{r}$ forks over $\emptyset$ because there are infinitely many
distinct choices of $a_{r}^{j}$ giving different curves. Any pair of the
curves $\ell_{j}=\langle a_{1},a_{2}\dots a_{r-1},a^{j}_{r}\rangle$ intersect
only in $a_{1},a_{2}\dots a_{r-1}$ since any further point of intersection
would make the curves identical. But then the formula $R(a_{1},a_{2},\dots
a_{r-1},x)$ forks over $\emptyset$ and $\operatorname{nmdeg}(p)=r-1$. ∎

######  Remark 5.3.26.

We compare our construction with similar construction of Evans [28] who wrote:

> By a $t-(v,k,\lambda)$ design we mean a set P of points, of cardinality v,
> together with a set $\mathcal{B}$ of blocks each of which is a k-subset of
> P, and which has the property that any set of t points is a subset of
> exactly $\lambda$ blocks. An automorphism of the design (P, B) is simply a
> permutation of P which preserves $\mathcal{B}$. It is well known [28] that
> if G is a group of automorphisms of the design (P, B) and $t\geq 2$ and
> $v,k,\lambda$ are finite, then the number of G-orbits on P is no greater
> than the number of G-orbits on $\mathcal{B}$: this is commonly referred to
> as ‘Block’s Lemma’.

[28] proves ‘For all ‘reasonable’ finite t, k and s we construct a
$t-(\aleph_{0},k,1)$ design and a group of automorphisms which is transitive
on blocks and has s orbits on points.’ We can rephrase our current result in
this language as:

######  Corollary 5.3.27.

Each model of cardinality $\kappa$ is an $r-(\kappa,\kappa,1)$ design and the
restriction to strongly minimal subset $P$, is an $r-(\kappa,k,1)$ design.
There are two orbits on points: $P$ and $\neg P$, the action is transitive on
blocks.

Evans’ structure is a $k$-Steiner system so more similar to [7]. The infinite
lines are necessary for the nmdeg result. By adding additional unary
predicates $P_{i}$ and letting line length in $P_{i}$ depend on $i$, we would
recover Evans’ result. But with one class of infinite lines.

##  6\. The Koponen conjecture

Koponen asks the following question ([56], [58], [59]):

######  Question 6.1.

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

Particularly, Koponen conjectures that the answer to the question of whether
such a theory is supersimple of finite rank is yes, proposing that it should
likely be possible to show this using the stability-theoretic techniques
developed for simple theories. In this section, we resolve Koponen’s
conjecture:

######  Theorem 6.2.

Let $T$ be a simple theory with quantifier elimination in a finite relational
language. Then $T$ is one-based, so in particular, supersimple of finite rank.

To show this, we prove a result of independent interest on the structure of
$\omega$-categorical simple theories.

######  Theorem 6.3.

Let $T$ be $\omega$-categorical and simple, but not supersimple. Then there is
a finite tuple $b$ such that for $i<\omega$, ($\emptyset$-)indiscernible
sequences $I^{i}=(b^{i}_{j})_{j<\omega}$ with $b^{i}_{0}=b$, so that
$I^{i}\not\equiv I^{j}$ for $i\neq j<\omega$.

Note that 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
$n<\omega$.121212Recall that a theory is $n$-ary if it has quantifier
elimination in a language with only $n$-ary relation symbols. Assuming $T$ as
in the theorem is not just simple, but supersimple, we will see that one proof
of the theorem is just a corollary of Fact 3.2.11 on $k$-linearity for
$\omega$-categorical supersimple theories, due to Tomašić and Wagner ([86]).
When $\mathrm{SU}(p)=1$ (i.e., in the case of [86] used in the finite-rank
case of the Koponen conjecture), that $k$-linearity of $p$ coincides with the
maximum value of $F_{Mb}(p\otimes p)$ being equal to $k$ was observed in
Corollary 3.2.9.

Another proof uses [86] in the case that $T$ is already supersimple of finite
rank, but directly uses $F_{Mb}$, in particular Corollary 3.2.20, to show that
if $T$ is supersimple it must have finite rank. The hardest new part of the
proof of this theorem is then proving that a simple theory with quantifier
elimination in a finite relational language must be supersimple.

We will need the following lemma on non-supersimple simple theories:

######  Lemma 6.4.

Let $T$ be simple, but not supersimple. Then there is a limit ordinal
$\lambda$, a finite tuple $a$, and sets $B_{\gamma}$, $\gamma<\lambda$, with
$B_{\gamma}\subset B_{\gamma^{\prime}}$ and $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{B_{\gamma}}B_{\gamma^{\prime}}$ for
$\gamma<\gamma^{\prime}<\lambda$, so that for
$B=\bigcup_{\lambda<\gamma}B_{\gamma}$,
$\mathrm{SU}(\mathrm{tp}(a/B))\in\mathrm{Ord}$.

###### Proof.

Suppose otherwise, so for any limit ordinal $\lambda$, finite tuple $a$, and
sets $B_{\gamma}$, $\gamma<\lambda$, with $B_{\gamma}\subset
B_{\gamma^{\prime}}$ and $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{B_{\gamma}}B_{\gamma^{\prime}}$ for
$\gamma<\gamma^{\prime}<\lambda$, for $B=\bigcup_{\lambda<\gamma}B_{\gamma}$,
$\mathrm{SU}(\mathrm{tp}(a/B))=\infty$. By transfinite induction, we will find
a finite tuple $a$ and sets $B_{\gamma}$, $\gamma\in\mathrm{Ord}$, with
$B_{\gamma}\subset B_{\gamma^{\prime}}$, $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{B_{\gamma}}B_{\gamma^{\prime}}$ and
$\mathrm{SU}(\mathrm{tp}(a/B_{\gamma}))=\infty$ for
$\gamma<\gamma^{\prime}\in\mathrm{Ord}$, contradicting simplicity. The base
case is just that $T$ is not supersimple. Suppose we have found $a$, and
$B_{\gamma^{\prime}}$ as desired for $\gamma^{\prime}\leq\gamma$. Then
$\mathrm{SU}(\mathrm{tp}(a/B_{\gamma}))=\infty$, so we can find $B_{\gamma+1}$
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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{B_{\gamma}}B_{\gamma+1}$ and
$\mathrm{SU}(\mathrm{tp}(a/B_{\gamma}))=\infty$, completing the successor
step. Finally, for the limit step, we must show that for $\lambda$ a limit
ordinal, if $B_{\gamma}\subset B_{\gamma^{\prime}}$ and
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{B_{\gamma}}B_{\gamma^{\prime}}$ for
$\gamma<\gamma^{\prime}<\lambda$, and
$B_{\lambda}=\bigcup_{\lambda<\gamma}B_{\gamma}$,
$\mathrm{SU}(\mathrm{tp}(a/B_{\lambda}))=\infty$; then $B_{\lambda}$ can be
chosen for stage $\lambda$ of the induction. But this was just the assumption
we made in supposing our conclusion was false. ∎

We now prove Theorem 6.3:

###### Proof.

Assume $T$ is simple and $\omega$-categorical, but not supersimple. Let $a$
and $B$ be as in Lemma 6.4; then
$\mathrm{SU}(\mathrm{tp}(a/B))\in\mathrm{Ord}$, but for any $B_{0}\subseteq B$
with $|B_{0}|<\omega$, $\mathrm{SU}(\mathrm{tp}(a/B))=\infty$. Let $|a|=k$.
Recalling Definition 3.2.14, we claim that it suffices to show that, for
arbitrarily large $n$, there exists a set $A_{n}\subset\mathbb{M}$ and type
$p_{n}\in S^{k}(A_{n})$ that is $(n+1)$-resolvable but not $n$-resolvable.
This suffices because, by the equivalence between (1), (2) and (2’) in
Proposition 3.2.16, a Morley sequence $J^{n}$ in $p_{n}$ over $A_{n}$ will be
$(n+1)$-resolvable but not $n$-resolvable (as an indiscernible sequence over
$\emptyset$), so $J^{n}\not\equiv J^{m}$ for $n\neq m$, giving us the
conclusion of Theorem 6.3 (by an automorphism, and either
$\omega$-categoricity, or the fact that the $p_{n}$ we get in the following
argument extend $\mathrm{tp}(a/\emptyset)$.)

Suppose that these $A_{n}$, $p_{n}$ do not exist for arbitrarily large $n$.
Then there is some $n<\omega$ such that for $A$ any set and $p\in S^{k}(A)$,
if $p$ is $m$-resolvable for some $m<\omega$, then it is $n$-resolvable. Let
$|x|=n$; then by $\omega$-categoricity, there is a formula $\varphi(a,x)$ so
that
$\varphi(a,\mathbb{M}^{n})=\\{c\in\mathbb{M}^{n}:\mathrm{SU}(\mathrm{tp}(a/c))=\infty\\}$,
and for any finite $B_{0}\subset B$ there is a formula
$\varphi^{B_{0}}(a,B_{0},x)$ so that
$\varphi^{B_{0}}(a,B_{0},\mathbb{M}^{n})=\\{c\in\mathbb{M}^{n}:a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{c}B_{0}\\}$.
We will show that the set

$\\{\varphi(a,x)\\}\cup\\{\varphi^{B_{0}}(a,B_{0},x):B_{0}\subset
B,|B_{0}|<\omega\\}$

is consistent. By compactness, for any finite collection $S$ of finite subsets
of $B$, we need only show

$\\{\varphi(a,x)\\}\cup\\{\varphi^{B_{0}}(a,B_{0},x):|B_{0}|\in S\\}$

is consistent. But by monotonicity of independence, $\varphi^{\cup S}(a,\cup
S,x)$ implies $\varphi^{B_{0}}(a,B_{0},x)$ for all $B_{0}\in S$, so we may
assume $|S|=1$, and it suffices to show that
$\\{\varphi(a,x)\wedge\varphi^{B_{0}}(a,B_{0},x)\\}$ is consistent for an
arbitrary finite $B_{0}\subseteq B$. Tautologically, the type
$p=\mathrm{tp}(a/B_{0})$ is $|B_{0}|$-resolvable, and therefore, by
assumption, $n$-resolvable. There is therefore some $c_{0}$ with $|c_{0}|\leq
n$ (enumerable as a tuple of length $n$) and $q\in S(B_{0}c_{0})$ with
$p\subseteq q$ that forks over neither $B_{0}$ or $c_{0}$; by an automorphism,
we may further choose $c_{0}$ and $q$ so that $a\models q$. Since
$\mathrm{SU}(p)=\infty$ and $q$ does not fork over $p$,
$\mathrm{SU}(q)=\infty$; therefore,
$\mathrm{SU}(q|_{c_{0}})=\mathrm{SU}(\mathrm{tp}(a/c_{0}))=\infty$. Moreover,
because $q$ does not fork over $c_{0}$, $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{c_{0}}B_{0}$,
so $c_{0}\models\\{\varphi(a,x)\wedge\varphi^{B_{0}}(a,B_{0},x)\\}$, showing
consistency of
$\\{\varphi(a,x)\\}\cup\\{\varphi^{B_{0}}(a,B_{0},x):B_{0}\subset
B,|B_{0}|<\omega\\}$. Now let $c$ realize this set; then
$\mathrm{SU}(a/c)=\infty$ 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 all $B_{0}\subset B$ with
$|B_{0}|<\omega$; by finite character of independence,
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{c}B$. Therefore,
$\mathrm{SU}(\mathrm{tp}(a/Bc))=\infty$, so
$\mathrm{SU}(\mathrm{tp}(a/B))=\infty$, a contradiction. This concludes the
proof of Theorem 6.3. ∎

We now prove Theorem 6.2. Each step of the proof will be an application of the
following observation:

######  Lemma 6.5.

If $T$ has quantifier elimination in a finite relational language, then there
is no tuple $b$ with $|b|<\omega$ and, for $i<\omega$,
($\emptyset$-)indiscernible sequences $I^{i}=(b^{i}_{j})_{j<\omega}$ with
$b^{i}_{0}=b$, so that $I^{i}\not\equiv I^{j}$ for $i\neq j<\omega$.

###### Proof.

We show that there are only finitely many types which
$\emptyset$-indiscernible sequences $I=\\{b_{i}\\}_{i<\omega}$, where
$|b_{0}|=b$, can realize over $\emptyset$, proving the lemma. By $n$-arity of
$T$ and indiscernibility of $I$, $\mathrm{tp}(b_{0}\ldots b_{n-1})$ determines
$\mathrm{tp}(I)$. But by $\omega$-categoricity, there are only finitely many
possibilities for $\mathrm{tp}(b_{0}\ldots b_{n-1})$.

∎

It is then immediate from Theorem 6.3 that, if $T$ is a simple theory with
quantifier elimination in a finite relational language, then $T$ is
supersimple. That if $T$ has quantifier elimination in a finite relational
language and is supersimple, $T$ is one-based, can be proven in two ways,
which we do next.

The first proof uses the following result of Koponen, which generalizes the
main result of [40]:

######  Fact 6.6.

(Theorem 3.1, [58]) Let $T$ be $\omega$-categorical and supersimple, and
suppose that forking is not geometricallly trivial (Definition 4.12). Then
there is a regular type over a finite set with a nontrivial pregeometry.

We claim that if $T$ has quantifier elimination in a finite relational
language and is supersimple, it suffices to show that every regular type over
a finite set is linear. Suppose we have shown this; then either forking in $T$
is trivial, or by the fact, there is a nontrivial linear regular type over a
finite set. But note the following result of [86] (see the proof of Theorem
5.5.4 of [85], for the case of $\mathrm{SU}(p)=1$):

######  Fact 6.7.

If $T$ is an $\omega$-categorical simple theory with a regular type over a
finite set that is either $k$-linear for $k>1$, or linear with nontrivial
pregeometry, there is an infinite group interpretable in $T$.

(For $k>1$, this is the main ingredient of the proof of Fact 3.2.11; however,
we only need to use $k=1$ directly here.) However, note also the following
result of MacPherson ([63]).

######  Fact 6.8.

A theory with quantifier elimination in a finite relational language cannot
interpret an infinite group.

Therefore, there can be no nontrivial linear regular type, so we are in the
case where forking in $T$ is trivial. (Note the similarity to the observation
directly below Fact 2.6 of [57], where the main result of [40] is used in
place of 6.6, and [78] is used in place of Fact 6.7.) However, Corollary 3.12
of [75], restated in Proposition 3.6 of [58], states that an omega-categorical
supersimple theory with trivial forking must have finite rank. Thus, we can
apply the observation between Fact 2.6 of [57]:

######  Fact 6.9.

Let $T$ be an $\omega$-categorical simple theory of finite rank. Then $T$ is
one-based if and only if forking is geometrically trivial in $T$.

So $T$ is one-based.

So it remains to show that every regular type over a finite set in $T$ is
linear. If a regular type $p(x)\in S(A)$, with $A$ finite, is not linear, then
by Corollary 3.2.12, $F_{Mb}(p\otimes p)>N$ for $p\otimes p$ some type
realized by two $A$-independent realizations of $p$, and arbitrarily large
$N$. (In fact, when $T$ is of finite rank, $F_{Mb}(p\otimes p)=\infty$ by
Proposition 4.2, or because the type $p$ produced in the proof in [57] of Fact
6.6 can be chosen to be Lascar, we can apply the Lascar case of 3.2.12 to get
$F_{Mb}(p\otimes p)=\infty$.) But recalling the definition of $F_{Mb}(p\otimes
p)$, this produces, for arbitrarily large $N$, an $A$-indiscernible sequence
$\\{a_{i}\\}_{i<\omega}$ with $|a_{i}|=2|x|$, any $N$ terms of which are
independent, but any $N+1$ terms of which are dependent, contradicting (recall
that $A$ is assumed to be finite) Lemma 6.5. (Note, however, that it is not
strictly necessary to use $F_{Mb}$ here: $p$ is pseudolinear by Fact 3.2.11,
so $p\otimes p$ has forking extensions $q\in S(B)$, $B\subseteq A$ with
canonical bases of unbounded $\mathrm{SU}_{p}$-rank, where $p\otimes p$ ranges
over types over $A$ of two independent realizations of $p$ over $A$. By taking
Morley sequences over $B$ in the extensions $q$, we get infinitely many
indiscernible sequences realizing different types over $A$, because
$\mathrm{SU}_{p}(\mathrm{Cb}(q))$ is an invariant of these indiscernible
sequences by Remark 3.2.13.) This gives the first proof of the supersimple
case of the Koponen conjecture, thereby proving Theorem 6.2.

In the second proof, we avoid having to use the full force of Fact 3.2.11 and
Corollary 3.2.12, restricting ourselves to the case where $\mathrm{SU}(p)=1$.
The case where $T$ is supersimple of finite rank is proved as before: If $T$
is not one-based, then by Fact 6.9, forking is not geometrically trivial, so
by Fact 6.6 produces a type $p$ over a finite set with $\mathrm{SU}(p)=1$ and
nontrivial pregeometry. Then by the rank-one case of Fact 6.7 (which can be
found in the proof of 5.5.4 of [86]), and Fact 6.8, $p$ cannot be linear; note
that, by the proof in [57] of Fact 6.6, the type $p$ in that fact can clearly
be chosen to be Lascar, so the main theorem of [78] will also give us an
infinite interpretable group. Since $p$ is not linear and has
$\mathrm{SU}(p)=1$, we can then apply Fact 3.2.12, but this time only the
$\mathrm{SU}$-rank $1$ case, to reach a contradiction as in the first proof.

In the case where $T$ is supersimple, it remains to show that $T$ is of finite
rank. But otherwise, there is a type $p$, such that $\mathrm{SU}(p)$ is a
limit ordinal. Then by Corollary 3.2.20, $F_{Mb}(p)=\infty$, again
contradicting Lemma 6.5.131313Observe that to get from the general supersimple
case to the finite-rank case, we now apply $F_{Mb}$ without referring to
canonical bases. However, note that by Corollary 3.2.9, in the case where
$\mathrm{SU}(p)=1$, $k$-linearity of $p$ is equivalent to the maximum value of
$F_{Mb}(p\otimes p)$, where $p\otimes p$ ranges over the types of two
independent realizations of $p$, being equal to $k$; in this sense, even the
first, finite-rank step step of the proof constitutes an application of
$F_{Mb}$.

We can generalize this question to $\mathrm{NSOP}_{1}$ theories (or even
further; see [68], [67].) The independence definition of being one-based (Fact
2.1.8) can be generalized to $\mathrm{NSOP}_{1}$ theories:

######  Definition 6.10.

Let $T$ be $\mathrm{NSOP}_{1}$. Then $T$ is _one-based_ 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}B$ implies (equivalently, is equivalent
to) $\mathrm{acl}^{eq}(AM)\cap\mathrm{acl}^{eq}(BM)\supsetneq M$.141414To be
certain that this definition extends the definition of being one-based in
simple theories, we should have used $\mathrm{bdd}$ instead of
$\mathrm{acl}^{eq}$. However, as related to us in a personal communication
with Nicholas Ramsey, there is no known $\mathrm{NSOP}_{1}$ that does not
eliminate hyperimaginaries, so in all known examples $\mathrm{acl}^{eq}$
coincides with $\mathrm{bdd}$.

We ask whether Koponen’s conjecture holds for all $\mathrm{NSOP}_{1}$
theories.

######  Question 6.11.

Let $T$ be $\mathrm{NSOP}_{1}$ and have quantifier elimination in a finite
relational language. Is $T$ one-based?

##  7\. Definability of Morley sequences and the simple Kim-forking conjecture

In the previous section, we characterized forking in a simple theory with
quantifier elimination in a finite relational language, and asked (Question
6.11) whether this characterization extends to Kim-forking in
$\mathrm{NSOP}_{1}$ theories with quantifier elimination in a finite
relational language. By Lemma 6.5, a more immediate consequence of an
$\mathrm{NSOP}_{1}$ theory having quantifier elimination in a finite
relational language (i.e., being $\omega$-catgorical and $n$-ary) is that
$F_{Mb}(p)$ is finite for $p$ a type over a finite set; here we extend
$F_{Mb}$ to Kim-independence:

######  Definition 7.1.

Let $T$ be $\mathrm{NSOP}_{1}$. Let $p(x)$ be a complete type over a model
$M$. Let $F_{Mb}(p)$ be the least $n<\omega$ so that, if
$\\{b_{i}\\}_{i<\omega}$ is an $M$-indiscernible sequence with $b_{i}\models
p(x)$ and $b_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}b_{<i}$ for all $i\leq n$, then
$\\{b_{i}\\}_{i<\omega}$ is an $\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}$-Morley sequence over $M$; if such $n$
does not exist, define $F_{Mb}(p)=\infty$.

However, in Section 4, we saw that it is possible for $F_{Mb}$ to be finite
for all types in a theory $T$, even a stable one, with more complicated
forking (i.e., $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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 just coincide with
$\mathrm{acl}(aC)\cap\mathrm{acl}(bC)=\mathrm{acl}(C)$). In this section, we
will show that when a type $p$ in an $\mathrm{NSOP}_{1}$ theory shares some
properties with types in theories eliminating quantifiers in a finite
relational language, including $F_{Mb}(p)<\omega$ or definability of
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}$-Morley sequences in $p$ (Definition 7.18
below), the Kim-forking extensions of $p$ must still be described by
generalizations of the _stable forking conjecture_. The stable forking
conjecture, due to Kim (see, e.g., [50] for the the definition of the “stable
forking property,” the conclusion of this conjecture) states:

######  Conjecture 7.2.

(Stable forking conjecture) Let $T$ be a simple $\mathcal{L}$-theory. 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}b$, there is a formula
$\varphi(x,c)\in\mathrm{tp}(a/Cb)$ (where $c\subset Cb$) which forks over $C$,
so that $\varphi(x,y)$ is a stable $\mathcal{L}$-formula.

This conjecture can be generalized to $\mathrm{NSOP}_{1}$ theories. Since an
$\mathrm{NSOP}_{1}$ theory is not necessarily simple, we can also demand only
a simple formula that witnesses Kim-forking, rather than a stable formula.

######  Question 7.3.

(Stable (or simple) Kim-forking-conjecture) Let $T$ be an $\mathrm{NSOP}_{1}$
$\mathcal{L}$-theory, and suppose $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}b$. Must there be a formula
$\varphi(x,b)\in\mathrm{tp}(a/Mb)$, with $\varphi(x,y)\in\mathcal{L}(M)$ so
that:

(a) $\varphi(x,b)$ Kim-forks over $M$.

(b) $\varphi(x,y)$ is stable (or simple, i.e. lacks the tree property) as an
$\mathcal{L}(M)$-formula?

For $\mathrm{NSOP}_{1}$ theories with existence (see [24]), this question is
easily modified to hold over sets.

######  Remark 7.4.

The stable Kim-forking conjecture is referred to as the weak stable Kim-
forking conjecture in [15], where Bossut observes that it is in fact necessary
to modify the conjecture so that $\varphi(x,y)$ ranges over the stable
formulas in $\mathcal{L}(M)$, rather than in $\mathcal{L}$: a theory
satisfying the latter, stronger statement (i.e., the conclusion of Conjecture
7.2, where any reference to forking is replaced with Kim-forking) is in fact
simple itself, so that statement is in fact false for strictly
$\mathrm{NSOP}_{1}$ theories. Similarly, the statement of the simple Kim-
forking conjecture must be modified so that $\varphi(x,y)$ ranges over the
simple formulas in $\mathcal{L}(M)$. Suppose that that $T$ is
$\mathrm{NSOP}_{1}$, and the conclusion of Conjecture 7.2 holds, but replacing
any reference to forking with Kim-forking and replacing stable with simple; we
show $T$ must be simple. Let $M\prec M^{\prime}\subset B$ and
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M^{\prime}}B$. Then by the assumption,
there is a formula $\varphi(x,b)\in\mathrm{tp}(a/B)$ that Kim-forks over
$M^{\prime}$ with $\varphi(x,y)$ a simple $\mathcal{L}$-formula. Then
$\varphi(x,b)$ divides over $M$, so because $\varphi(x,y)$ is a simple
$\mathcal{L}$-formula, $\varphi(x,b)$ Kim-divides over $M$, and
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}B$. So $\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 5.71527pt\scriptscriptstyle{}}}^{K}$ is base-
monotone in the sense of Proposition 8.8 of [48], so $T$ is simple by that
proposition.

In this section, we will focus on the simple Kim-forking conjecture. There are
other ways to weaken the stable Kim-forking conjecture besides requiring only
a simple formula, rather than a stable formula, to witness Kim-forking. For
example, one can assert that the forking (or Kim-forking) relation itself,
which is not necessarily definable, lacks the order property or tree property.
In other words, we are interested in showing the stable forking conjecture, or
stable (simple) Kim-forking conjecture, is true up to definability of the
independence relation. Depending on how this assertion is formulated, this
will weaken the stable forking conjecture or the stable (simple) Kim-forking
conjecture.

For example, in Theorem 4.1 of [73], Palacin and Wagner prove stability of the
forking relation, which is not a priori definable, in supersimple CM-trivial
theories. More precisely, for the forking relation

$R(x,y_{1},y_{2}):=x\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{y_{1}}y_{2},$

which is not necessarily definable, [73] show that $R(x,y_{1}y_{2})$ is
stable, or lacks the order property: there are no
$\\{a^{i},b_{1}^{i}b_{2}^{i}\\}_{i<\omega}$ so that $\models
R(a^{i},b_{1}^{j}b_{2}^{j})$ exactly when $i<j$.

######  Remark 7.5.

Another weaker formulation of stability of
$R(x,y_{1},y_{2})=:x\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{y_{1}}y_{2}$ is to require that there are no
$\\{a^{i},b_{1}^{i}b_{2}^{i}\\}_{i<\kappa}$, for $\kappa$ large enough to
apply the Erdős-Rado theorem, so that $\models R(a^{i},b_{1}^{j}b_{2}^{j})$
exactly when $i<j$. Similarly, we can formulate stability for
$R(x,y)=:x\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}y$, where $M$ is some fixed model, so
that $R(x,y)$ is defined to be stable if there are no
$\\{a^{i}b^{i}\\}_{i<\kappa}$ so that $\models R(a^{i},b^{j})$ exactly when
$i<j$. Under this formulation, stability of the forking relation, or Kim-
forking relation, itself is weaker than the stable forking conjecture. For
example, in the case where $R(x,y)$ is the Kim-forking relation, if the
conclusion of the stable Kim-forking conjecture holds we can show that there
are no $\\{a^{i}b^{i}\\}_{i<\kappa}$ so that $R(a^{i},b^{j})$ exactly when
$i<j$. If such $\\{a^{i}b^{i}\\}_{i<\kappa}$ existed, by the Erdős-Rado
theorem we can assume $\\{a^{i}b^{i}\\}_{i<\kappa}$ is indiscernible over $M$.
But by the assumption, whenever $i<j$, there is some formula
$\varphi(x,b^{j})\in\mathrm{tp}(a^{i}/Mb^{j})$, which Kim-forks over $M$ so
that $\varphi(x,y)$ is stable as a $\mathcal{L}(M)$-formula; by
indiscernibility, for $i<j$, $\varphi(x,y)$ does not depend on $i,j$. Then
$\models\varphi(a^{i},b^{j})$ exactly when $i<j$, contradicting stability of
$\varphi(x,y)$.

######  Remark 7.6.

On the other hand, the proof of Corollary 4.4 of [73] says that, if $T$ is
$\omega$-categorical and supersimple, and the forking relation is assumed to
be stable (even in the weaker sense described in the previous remark), the
conclusion of the stable forking conjecture holds. The proof from [73] is as
follows: let $a$, $b$ be finite tuples 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}b$ and, by supersimplicity, find finite
$C_{0}\subset C$ so that $ab\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C_{0}}C$. By $\omega$-categoricity, the
relation $R(x,y_{1}y_{2})=:x\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{y_{1}}y_{2}$ is definable by a formula
$\varphi(x,y_{1}y_{2})$, and since $R(x,y_{1}y_{2})$ is a stable relation,
$\varphi(x,y_{1}y_{2})$ is a stable formula. But
$\varphi(x,C_{0}b)\in\mathrm{tp}(a/Cb)$, and $\varphi(x,C_{0}b)$ can be seen
to fork over $C$, since it forks over $C_{0}$ 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$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{C_{0}}C$.

Moreover, when the forking relation is assumed to be stable, it is even easier
to see that when $T$ is an $\omega$-categorical simple theory, the stable
forking conjecture is true over finite sets: the stable formula witnessing
this is just the forking relation itself. When $T$ is $\mathrm{NSOP}_{1}$ with
existence (so forking is defined over all sets, including finite sets), and
the Kim-forking relation is stable, the stable Kim-forking conjecture is true
over finite sets.

Having discussed what it means for the forking or Kim-forking relation to lack
the order property, we must make precise what it means for the Kim-forking
relation over a model (with variables for sets, say, of size less than some
given bound) to lack the tree property, that is, to be simple as a relation.
To this end, we give a notation for consistency and inconsistency of relations
that are not necessarily definable, in order to extend the definition of
simplicity to these relations.

######  Definition 7.7.

Let $S$ and $T$ be arbitrary sets. Let $R\subseteq S\times T$ be a relation.
Let $\\{t_{i}\\}_{i\in I}\subseteq T$ be an indexed subset of $T$. Then we use
the notation that $\\{R(x,t_{i})\\}_{i\in I}$ is consistent to mean that there
is some $s\in S$ so that $(s,t_{i})\in R$ for all $i\in I$. For $k<\omega$, we
use the notation that $\\{R(x,t_{i})\\}_{i\in I}$ is $k$-inconsistent to mean
that for every $I_{0}\subseteq I$ with $|I_{0}|\leq k$, it is not the case
that $\\{R(x,t_{i})\\}_{i\in I}$ is consistent.

######  Definition 7.8.

Let $\kappa_{1}$ be infinite cardinals, $S$ and $T$ arbitrary sets, and
$R\subseteq S\times T$ a relation. Then $R$ is
$(\kappa_{1},\kappa_{2})$-simple if there is no $k<\omega$ and tree-indexed
subset $\\{t_{\eta}\\}_{\eta\in\kappa_{2}^{<\kappa_{1}}}\subseteq T$, so that,
for all $\sigma\in\kappa_{2}^{\kappa_{1}}$,
$\\{R(x,t_{\sigma|_{\lambda}})\\}_{\lambda<\kappa_{1}}$ is consistent, and for
$\eta\in\kappa_{2}^{<\kappa_{1}}$,
$\\{R(x,t_{\eta\smallfrown\langle\lambda\rangle})\\}_{\lambda<\kappa_{2}}$ is
$k$-inconsistent.

Note that if $S=\mathbb{M}^{n}$, $T=\mathbb{M}^{m}$ and
$R=:\\{(a,b):\mathbb{M}\models\varphi(a,b)\\}$ (where $\varphi(x,y)$ is an
$\mathcal{L}$-formula with $|x|=n$ and $|y|=m$) is an $\mathcal{L}$-definable
relation, then for $\kappa_{1},\kappa_{2}$ any infinite cardinals, $R$ is
$(\kappa_{1},\kappa_{2})$-simple if and only if $\varphi(x,y)$ is a simple
formula. That is, Definition 7.8 extends the usual notion of simplicity for
formulas.

The main tool we will use is the following theorem of Kaplan and Ramsey
(Corollary 5.4 of [49]), extended to arbitrary sets in $\mathrm{NSOP}_{1}$
theories with existence by Chernikov, Kim and Ramsey in [4] (the hard
direction, from (b) to (a), is Theorem 2.8 of [4], and the easy direction is
as in the easy direction of Corollary 5.4 of [49].)

######  Fact 7.9.

(Kaplan and Ramsey, [49], and Chernikov, Kim and Ramsey, [4])

Suppose $T$ is $\mathrm{NSOP}_{1}$. For any $M\prec\mathbb{M}$ and
$b\subset\mathbb{M}$, if $\\{b_{i}\\}_{i<\omega}$ is an $M$-indiscernible
sequence with $b_{0}=b$, then the following are equivalent:

(a) For all formulas $\varphi(x,b)$ that Kim-divide over $M$,
$\\{\varphi(x,b_{i})\\}_{i<\omega}$ is inconsistent.

(b) The sequence $\\{b_{i}\\}_{i<\omega}$ is an $\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 5.71527pt\scriptscriptstyle{}}}^{K}$-Morley
sequence over $M$: $b_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}^{K}b_{<i}$ for $i<\omega$.

If $T$ is $\mathrm{NSOP}_{1}$ with existence, then the above holds even when
$M$ is assumed only to be a (small) subset of $\mathbb{M}$, and not
necessarily a model.

If we allow the Kim-forking relation $R$ to have infinitely many variables,
then a global analogue of the simple Kim-forking conjecture turns out to hold
in all $\mathrm{NSOP}_{1}$ theories. Contrast this with the Theorem 4.1 of
[73], where the forking relation with finitely many variables is shown to be
stable (in the sense of sequences of length $\omega$), but $T$ is required to
be not just simple but supersimple and CM-trivial.

######  Theorem 7.10.

Let $T$ be $\mathrm{NSOP}_{1}$. Let $M\prec\mathbb{M}$, and let
$\kappa=2^{|T|+|M|}$. Let $S_{1}=:\\{a\subset\mathbb{M}:|a|<\kappa\\}$, and
let $S_{2}=\mathbb{M}^{n}$ for some fixed $n$. Let $R=:\\{(a,b)\in S_{1}\times
S_{2}:a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}b\\}$. Then $R$, as a relation on
$S_{1}\times S_{2}$, is $(\kappa,\omega)$-simple.

In this proof and in the proofs of Theorems 7.14 and 7.20, it may have been
possible to shorten the proofs using results on indiscernible trees; see, e.g.
[79] or [84] for an overview, as well as some new results in this area. To
keep the proofs self-contained, we instead give direct proofs.

###### Proof.

Suppose not, and let $\\{b_{\eta}\\}_{\omega^{<\kappa}}$ witness this for some
degree of inconsistency $k<\omega$. We must account for the notational
subtlety about limit ordinals pointed out in Remark 5.8 of [48], so let
$\mathrm{Succ}(\kappa)$ denote the successor ordinals, counting $0$, less than
$\kappa$. By transfinite induction on $\kappa^{\prime}$, we construct, for
$\kappa^{\prime}\leq\kappa$, some
$\sigma_{\kappa^{\prime}}\in\omega^{\kappa^{\prime}}$ so that, for all
$\lambda\in\mathrm{Succ}(\kappa^{\prime})$,
$b_{\sigma_{\kappa^{\prime}}|\lambda}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}^{K}\\{b_{\sigma_{\kappa^{\prime}}|\lambda^{\prime}}\\}_{\lambda^{\prime}\in\mathrm{Succ}(\lambda)}$,
and so that $\sigma_{\kappa^{\prime\prime}}\subset\sigma_{\kappa^{\prime}}$
for all $\kappa^{\prime\prime}<\kappa^{\prime}$. The case for
$\kappa^{\prime}=0$ and $\kappa^{\prime}$ a limit ordinal is clear, so it
remains to assume $\sigma_{\kappa^{\prime}}$ already constructed and find some
$m<\omega$ so that
$\sigma_{\kappa^{\prime}+1}=:\sigma_{\kappa^{\prime}}\smallfrown\langle
m\rangle$ works. By the induction hypothesis, it suffices to find some
$m<\omega$ so that $b_{\sigma_{\kappa^{\prime}}\smallfrown\langle
m\rangle}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}^{K}\\{b_{\sigma_{\kappa^{\prime}}|\lambda^{\prime}}\\}_{\lambda^{\prime}\in\mathrm{Succ}(\kappa^{\prime})}$.
Suppose otherwise, so by symmetry,
$\\{b_{\sigma_{\kappa^{\prime}}|\lambda^{\prime}}\\}_{\lambda^{\prime}\in\mathrm{Succ}(\kappa^{\prime})}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}^{K}b_{\sigma_{\kappa^{\prime}}\smallfrown\langle
m\rangle}$ for all $m<\omega$. Then
$a=\cup\\{b_{\sigma_{\kappa^{\prime}}|\lambda^{\prime}}\\}_{\lambda^{\prime}\in\mathrm{Succ}(\kappa)}$
has cardinality less than $\kappa$, so it witnesses that
$\\{R(x,b_{\eta\smallfrown\langle m\rangle})\\}_{m<\omega}$ is consistent,
when $\eta=\sigma_{\kappa^{\prime}}$. In particular,
$\\{R(x,b_{\eta\smallfrown\langle m\rangle})\\}_{m<\omega}$ cannot be
$k$-inconsistent, contradicting that
$\\{b_{\eta}\\}_{\eta\in\omega^{<\kappa}}$ is a witness to the failure of
$(\kappa,\omega)$-simplicity of $R$.

This completes the construction, so let $\sigma=\sigma_{\kappa}$. Then
$\\{R(x,b_{\sigma|_{\lambda}})\\}_{\lambda\in\mathrm{Succ}(\kappa)}$ is
consistent. Reindexing, we get a sequence $\\{b_{i}\\}_{i<\kappa}$ so that
$\\{R(x,b_{i})\\}_{i<\kappa}$ is consistent, and
$b_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}b_{<i}$ for $i<\kappa$. So some
$a\subseteq\mathbb{M}$, with $|a|<\kappa$, witnesses the consistency of
$\\{R(x,b_{i})\\}_{i<\kappa}$, meaning that $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}^{K}b_{i}$ for $i<\kappa$. This contradicts
the following claim, essentially Lemma C.7 of [60]:

######  Claim 7.11.

Let $|a|<\kappa$. Then there is no sequence $\\{b_{i}\\}_{i<\kappa}$, with
$b_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}b_{<i}$ for $i<\kappa$, and
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}^{K}b_{i}$ for $i<\kappa$.

###### Proof.

First, we may assume, by choice of $\kappa$, that $b_{i}\equiv_{M}b_{j}$ for
$i<j<\kappa$. Fix an enumeration $a=\\{a_{i}\\}_{i<\lambda}$ of $a$, where
$\lambda<\kappa$, and let $\overline{x}=\\{x_{i}\\}_{i<\lambda}$; let
$\overline{y}=\\{y_{i}\\}_{i<n}$. For each $i<\kappa$, let
$\varphi_{i}(\overline{x},b_{i})\in\mathrm{tp}(a/Mb_{i})$ Kim-fork over $M$.
By choice of $\kappa$, and the pigeonhole principle, we can assume that
$\varphi(\overline{x},\overline{y})=:\varphi_{i}(\overline{x},\overline{y})$
does not depend on $i$. We need the following subclaim, which is folklore:

######  Subclaim 7.12.

The set of sequences $\\{b^{\prime}_{i}\\}_{i<\kappa}$ with
$b^{\prime}_{i}\models p(\overline{y})=:\mathrm{tp}(b_{0}/M)$ and
$b^{\prime}_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}b^{\prime}_{<i}$ for $i<\kappa$ is
type-definable over $M$.

###### Proof.

By symmetry, this is the set of sequences $\\{b^{\prime}_{i}\\}_{i<\kappa}$
with $b^{\prime}_{i}\models p(\overline{y})$ and
$b^{\prime}_{<i}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}b^{\prime}_{i}$ for $i<\kappa$. Let $S$
be the set of formulas $\psi(\tilde{x},\overline{y})$ with parameters in $M$,
where $\tilde{x}=\\{x_{i}\\}_{i<\omega}$, so that $\psi(\tilde{x},b)$ Kim-
forks over $M$ for some (any) $b\models p(\overline{y})$. Then the above set
of sequences is just the set of sequences $\\{b^{\prime}_{i}\\}_{i<\kappa}$
with $b^{\prime}_{i}\models p(\overline{y})$ and no tuple from
$b^{\prime}_{<i}$ satisfying $\psi(\tilde{x},b^{\prime}_{i})$ for any
$\psi(\tilde{x},\overline{y})$ belonging to $S$; we easily see this is type-
definable. ∎

Find an $Ma$-indiscernible, and a fortiori $M$-indiscernible, sequence
$\\{b^{\prime}_{i}\\}_{i<\omega}$ with the same EM-type as
$\\{b_{i}\\}_{i<\kappa}$ over $Ma$; by the subclaim, this will be an
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}$-Morley sequence over $M$, and
$a\models\\{\varphi(\overline{x},b^{\prime}_{i})\\}_{i<\omega}$. So
$\\{\varphi(\overline{x},b^{\prime}_{i})\\}_{i<\omega}$ is consistent,
contradicting the direction (b) $\Rightarrow$ (a) of 7.9.

∎

∎

######  Remark 7.13.

The only place in the above where we specifically needed $\kappa=2^{|T|+|M|}$
is in assuming that $b_{i}\equiv_{M}b_{j}$ for $i<j<\kappa$. If we had made
$S_{2}$ the set of realizations of some complete $n$-type over $M$, we could
have chosen $\kappa=(|T|+|M|)^{+}$.

The previous result, while true of all $\mathrm{NSOP}_{1}$ theories, required
infinitely many variables $x$ in the Kim-independence relation
$x\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}y$; indeed, for
$(\omega,\kappa)$-simplicity, $x$ had to be a variable ranging over all sets
of size $\kappa$. This differs from [73], where all variables are finite. Here
we show that, when $y$ is a variable for a realization of $p(y)\in S(M)$ where
$F_{Mb}(p)<\infty$ (Definition 7.1), our results extend to a finitary Kim-
independence relation over a model. (Our arguments also give us a version, for
$x$ ranging over finite sets, of the previous theorem on general
$\mathrm{NSOP}_{1}$ theories.)

######  Theorem 7.14.

Let $T$ be $\mathrm{NSOP}_{1}$, and let $\kappa=(|M|+|T|)^{+}$. Fix a complete
$n$-type $p(y)$ over a model $M$. Let $F_{Mb}(p)\leq m<\omega+1$; if
$m<\omega$, let $S_{1}=\mathbb{M}^{n(m-1)}$, and if $m=\omega$, let
$S_{1}=\\{a\subseteq\mathbb{M}:|a|<\omega\\}$. Let
$S_{2}=\\{b\in\mathbb{M}^{n}:b\models p(y)\\}$, and let $R=:\\{(a,b)\in
S_{1}\times S_{2}:a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}b\\}$. Then $R$, as a relation on
$S_{1}\times S_{2}$, is $(\kappa,\kappa)$-simple.

###### Proof.

Suppose not, and let $\\{b_{\eta}\\}_{\omega^{<\kappa}}$ witness this for some
degree of inconsistency $k<\omega$. By transfinite induction on
$\kappa^{\prime}$, we construct, for $\kappa^{\prime}\leq\kappa$, some
$\sigma_{\kappa^{\prime}}\in\kappa^{\kappa^{\prime}}$ so that, for all
$\lambda\in\mathrm{Succ}(\kappa^{\prime})$,
$b_{\sigma_{\kappa^{\prime}}|\lambda}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}^{K}\\{b_{\sigma_{\kappa^{\prime}}|\lambda^{\prime}}\\}_{\lambda^{\prime}\in
I_{0}}$ for every $I_{0}\subset\mathrm{Succ}(\kappa^{\prime})$ with
$|I_{0}|<m$. Similarly to before, it remains to assume
$\sigma_{\kappa^{\prime}}$ already constructed and find some $\gamma<\kappa$
so that
$\sigma_{\kappa^{\prime}+1}=:\sigma_{\kappa^{\prime}}\smallfrown\langle\gamma\rangle$
works. By the induction hypothesis, it suffices to find some $\gamma<\kappa$
so that
$b_{\sigma_{\kappa^{\prime}}\smallfrown\langle\gamma\rangle}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}^{K}\\{b_{\sigma_{\kappa^{\prime}}|\lambda^{\prime}}\\}_{\lambda^{\prime}\in
I_{0}}$ for every $I_{0}\subset\mathrm{Succ}(\kappa^{\prime})$ with
$|I_{0}|<m$. Suppose otherwise, so by symmetry, for all $\gamma<\kappa$, there
is some $I_{0}\subset\mathrm{Succ}(\kappa^{\prime})$ with $|I_{0}|<m$ so that
$\\{b_{\sigma_{\kappa^{\prime}}|\lambda^{\prime}}\\}_{\lambda^{\prime}\in
I_{0}}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}^{K}b_{\sigma_{\kappa^{\prime}}\smallfrown\langle\gamma\rangle}$.
By the pigeonhole principle and choice of $\kappa$, there is some
$I_{0}\subset\mathrm{Succ}(\kappa^{\prime})$ with $|I_{0}|<m$ so that there is
an infinite set $I\subseteq\kappa$ so that, for every $\gamma\in I$,
$\\{b_{\sigma_{\kappa^{\prime}}|\lambda^{\prime}}\\}_{\lambda^{\prime}\in
I_{0}}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}^{K}b_{\sigma_{\kappa^{\prime}}\smallfrown\langle\gamma\rangle}$.
Then
$\cup\\{b_{\sigma_{\kappa^{\prime}}|\lambda^{\prime}}\\}_{\lambda^{\prime}\in
I_{0}}$ can be indexed as a tuple in $S_{1}$, so it witnesses that
$\\{R(x,b_{\eta\smallfrown\langle\gamma\rangle})\\}_{\gamma\in I}$ is
consistent, for $\eta=\sigma_{\kappa^{\prime}}$ In particular,
$\\{R(x,b_{\eta\smallfrown\langle\gamma\rangle})\\}_{\gamma<\kappa}$ cannot be
$k$-inconsistent.

This completes the construction, so let $\sigma=\sigma_{\kappa}$. Then
$\\{R(x,b_{\sigma|_{\lambda}})\\}_{\lambda\in\mathrm{Succ}(\kappa)}$ is