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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}\\{b_{i}\\}_{i\in I_{0}}$ for every
$i<\kappa$ and $I_{0}\subset[i]$ with $|I_{0}|<m$. If $m=\omega$ then by the
finite character 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}$, this just says
$b_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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 every $i<\kappa$. Again,
there is some $a\subseteq\mathbb{M}$, with $|a|<\kappa$ (indeed,
$|a|\leq(m-1)n$), which 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 $i<\kappa$. If $m=\omega$,
we proceed exactly as in the above. In case $m<\omega$, a contradiction
follows from the following version of Claim 7.11:

######  Claim 7.15.

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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}\\{b_{i}\\}_{i\in I_{0}}$ for every
$i<\kappa$ and $I_{0}\subset[i]$ with $|I_{0}|<m$, 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}^{K}b_{i}$ for $i<\kappa$.

###### Proof.

That $b_{i}\equiv_{M}b_{j}$ for $i<j<\kappa$ is now by definition on $S_{2}$.
Again, we may assume there is some formula
$\varphi(\overline{x},\overline{y})$ so that
$a\models\\{\varphi(x,b_{i})\\}_{i<\kappa}$ and $\varphi(x,b)$ Kim-forks over
$M$ for $b\models p(\overline{y})$. The proof of the following is a
straightforward generalization of the proof of Subclaim 7.12:

######  Subclaim 7.16.

The set of sequences $\\{b^{\prime}_{i}\\}_{i<\kappa}$ with
$b^{\prime}_{i}\models p(\overline{y})=:\mathrm{tp}(b_{0}/M)$, and 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}\\{b_{i}\\}_{i\in I_{0}}$ for every
$i<\kappa$ and $I_{0}\subset[i]$ with $|I_{0}|<m$, is type-definable over $M$.

Again, extract an $Ma$-indiscernible sequence
$\\{b^{\prime}_{i}\\}_{i<\omega}$ sharing the same EM-type as
$\\{b_{i}\\}_{i<\kappa}$ over $Ma$. By the subclaim,
$\\{b^{\prime}_{i}\\}_{i<\omega}$ will in particular be an $M$-indiscernible
sequence with $b^{\prime}_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}\\{b^{\prime}_{i}\\}_{i\in I_{0}}$ for
every $i<\omega$ and $I_{0}\subset[i]$ with $|I_{0}|<m$. Because
$F_{Mb}(p)\leq m$, $\\{b^{\prime}_{i}\\}_{i<\omega}$ 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 $M$. Conclude as in
the proof of Claim 7.11.

∎

∎

######  Remark 7.17.

The proof of the general part of Theorem 7.14 should also hold when the
argument $x$ of $R(x,y)$ is allowed to range over, for example, countable
sets. While in all $\mathrm{NSOP}_{1}$ theories, 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}y$ can then be seen by Theorems 7.10
and 7.14 to lack the $(\kappa,\kappa)$-tree property for sufficiently large
$\kappa$ as long as $x$ is a variable for an infinite or unbounded finite set,
when $x$ and $y$ are both finite sets of variables, it is unknown in general
whether the relation $x\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}y$ is $(\kappa,\kappa)$-simple for large
enough $\kappa$. However, like in Remark 7.5, we do see that
$(\kappa,\kappa)$-simplicity of the relation
$R(x,y)=:\\{a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}b,a\in\mathbb{M}^{m},b\in\mathbb{M}^{n}\\}$,
for $m$, $n$ finite and for large enough $\kappa$, does follow from the stable
Kim-forking conjecture itself: when $\kappa$ is large enough, if there is a
tree witnessing failure of the $(\kappa,\kappa)$-tree property, if the simple
Kim-forking conjecture is true we can find a subtree where the consistency of
the paths is witnessed by a single simple $\mathcal{L}(M)$-formula
$\varphi(x,y)$, but this subtree will witness that $\varphi(x,y)$ has the tree
property, a contradiction. (To find this subtree, we can, say, choose an
$\omega^{|T|^{+}}$-subtree inductively so that all paths have the same type,
then use the fact that the consistency of this path will involve the same
formula appearing infinitely many times to get a further subtree.) So as with
stability, $(\kappa,\kappa)$-simplicity of the Kim-independence relation gives
a weakening of the simple Kim-forking conjecture. Moreover, when $T$ is
$\omega$-categorical and $\mathrm{NSOP}_{1}$ with existence (so, again, Kim-
forking is defined over finite sets), $(\omega,\omega)$-simplicity of the
$R(x,y)=:\\{a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{C}b,a\in\mathbb{C}^{m},b\in\mathbb{M}^{n}\\}$
for $m,n$ finite is equivalent to the simple Kim-forking cojecture over $C$,
when $C$ is a finite set.

So far, we have discussed analogues of the simple Kim-forking conjecture for
the non-definable relation of Kim-forking. This leads to ask: what about the
simple Kim-forking conjecture itself? We will will show that, in
$\mathrm{NSOP}_{1}$ theories with existence, when $\mathrm{tp}(b/C)$ satisfies
the property that the notion of 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 in this type is
definable, forking with $b$ over $C$ satisfies the simple Kim-forking
conjecture. By Example 7.22 below, we will see that this gives us some
nontrivial instances of the simple Kim-forking conjecture. The following
property of a type $p$ strengthens the property that $F_{Mb}(p)$ is finite:

######  Definition 7.18.

Let $T$ be $\mathrm{NSOP}_{1}$ with nonforking existence, and let
$C\subseteq\mathbb{M}$ be a (small) set, and let $p(x)$ be a complete type
over $C$. Then $p(x)$ has definable $\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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 if the set 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 $\\{b_{i}\\}_{i<\omega}$
over $C$ with $b_{i}\models p(x)$ is definable over $C$ relative to the
(necessarily type-definable) set of indiscernible sequences
$\\{b_{i}\\}_{i<\omega}$ over $C$ with $b_{i}\models p(x)$. (That is, for
$S_{1}$ the set 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 $\\{b_{i}\\}_{i<\omega}$
over $C$ with $b_{i}\models p(x)$, and $S_{2}$ the set of $C$-indiscernible
sequences $\\{b_{i}\\}_{i<\omega}$ with $b_{i}\models p(x)$, there is a
formula $\varphi(x_{1},\ldots x_{n})$ with parameters in $C$, with
$|x_{i}|=|b_{i}|=k$, so that
$S_{1}=S_{2}\cap\\{\\{b_{i}\\}_{i<\omega}\in(\mathbb{M}^{k})^{\omega}:\mathbb{M}\models\varphi(b_{1},\ldots,b_{n})\\}$.)

######  Remark 7.19.

To see that if $p\in S(C)$ has definable $\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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, then $F_{Mb}(p)<\infty$, assume the former. For a $C$-indiscernible
sequence $I=\\{a_{i}\\}_{i<\omega}$, $a_{i}\models p(x)$ to 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 is equivalent to it being
$n$-independent for all $n<\omega$. But by the proof of Claim 7.12, being
$n$-independent is type-definable relative to $I$ being an indiscernible
sequence over $C$. So by definable $\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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 and compactness, there
is some $n<\omega$ so that if $I$ is $n$-independent, it 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, and therefore
$F_{Mb}(p)<\infty$.

We obtain an instance of the simple Kim-forking conjecture when, for
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}b$, $\mathrm{tp}(b/C)$ has definable
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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 and $a$ is given as a
tuple enumerated by a sufficiently long finite sequence (thereby giving us
enough variables to build the simple formula.)

######  Theorem 7.20.

Let $T$ be $\mathrm{NSOP}_{1}$ with existence. Let $C,b\subset M$ be such that
$p(y)=:\mathrm{tp}(b/C)$ has definable $\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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, and suppose that either $p(y)$ is isolated or $T$ is low. Then
there is some $n<\omega$ so that, for any $a=\\{a_{i}\\}_{i\in I}$ where
$|I|\geq n$ (with the $a_{i}$ allowed to repeat), and such 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{C}b$, there is some formula
$\varphi(x,b)\in\mathrm{tp}(a/Cb)$ that Kim-forks over $C$, where
$\varphi(x,y)$ is a formula with parameters in $C$ which is simple as a
formula of $\mathcal{L}(C)$.

###### Proof.

We first show that if there is a formula
$\tilde{\varphi}(x,b)\in\mathrm{tp}(a/Cb)$, Kim-forking over $C$, so that

(*) there is no $k<\omega$ and tree-indexed set
$\\{b_{\eta}\\}_{\eta\in\omega^{<\omega}}$, with $b_{\eta}\models p(y)$ for
all $\eta\in\omega^{<\omega}$, so that for all $\sigma\in\omega^{\omega}$,
$\\{\tilde{\varphi}(x,b_{\sigma|_{i}})\\}_{i<\omega}$ is consistent, but for
all $\eta\in\omega^{<\omega}$, $\\{\tilde{\varphi}(x,b_{\eta\smallfrown\langle
i\rangle})\\}_{i<\omega}$ is $k$-inconsistent.

then there is a formula $\varphi^{\prime}(x,b)\in\mathrm{tp}(a/Cb)$, Kim-
forking over $C$, so that $\varphi^{\prime}(x,y)$ is a formula with parameters
in $C$ that is simple as a formula of $L(C)$. If $p(y)$ is isolated, then
$\varphi^{\prime}(x,b)=:\tilde{\varphi}(x,b)\wedge p(b)$ will work. If $T$ is
low, then there is some $k$ so that if $\\{c_{i}\\}_{i<\omega}$ is any
indiscernible sequence, and $\\{\tilde{\varphi}(x,c_{i})\\}_{i<\omega}$ is
$k$-inconsistent, then it is inconsistent. Then for
$\\{c_{\eta}\\}_{\eta\in\omega^{<\omega}}$ to witness failure of simplicity
for $\tilde{\varphi}(x,y)$ as a formula of $L(C)$ means that for all
$\sigma\in\omega^{\omega}$,
$\\{\tilde{\varphi}(x,c_{\sigma|_{i}})\\}_{i<\omega}$ is consistent, but for
all $\eta\in\omega^{<\omega}$, $\\{\tilde{\varphi}(x,c_{\eta\smallfrown\langle
i\rangle})\\}_{i<\omega}$ is $k$-inconsistent. The property $(*)$ says that
$c_{\eta}\models p(y)$ for all $\eta\in\omega^{<\omega}$ implies that
$\\{c_{\eta}\\}_{\eta\in\omega^{<\omega}}$ does not witness the failure of
simplicity of $\tilde{\varphi}(x,y)$, but we see now that to witness the
failure of simplicity of $\tilde{\varphi}(x,y)$ is type-definable over $C$. So
by compactness, there is some $\rho(y)\in p(y)$ so that
$c_{\eta}\models\rho(y)$ for all $\eta\in\omega^{<\omega}$ implies that
$\\{c_{\eta}\\}_{\eta\in\omega^{<\omega}}$ does not witness the failure of
simplicity of $\tilde{\varphi}(x,y)$. Then
$\varphi^{\prime}(x,b)=:\tilde{\varphi}(x,b)\wedge\rho(b)$ will work.

As in the proof of Subclaim 7.12, let $S$ be the set of formulas
$\varphi(x,y)$ with parameters in $C$ so that $\varphi(x,b)$ Kim-divides over
$C$; then an indiscernible sequence $\\{b_{i}\\}_{i<\omega}$ over $C$, with
$b_{i}\models p(y)$, 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 $C$ if and only if,
for all $i<\omega$, no tuple from $b_{<i}$ satisfies $\varphi(x,b_{i})$ for
$\varphi(x,y)\in S$. So by 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 and compactness, there is some finite $x$ and finite subset
$\\{\varphi_{j}(x,y)\\}_{j<m}\subset S$ so that an indiscernible sequence
$\\{b_{i}\\}_{i<\omega}$ over $C$, with $b_{i}\models p(y)$, 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 $C$ if and only if,
for all $i<\omega$, no tuple from $b_{<i}$ satisfies $\varphi_{j}(x,b_{i})$
for all $j<m$.

Let $\psi(x,y)=\bigvee_{i<m}\varphi_{i}(x,y)$. Let $n=|x|$. Let
$a=\\{a_{i}\\}_{i\in I}$ where $|I|\geq n$ (with the $a_{i}$ allowed to
repeat), and 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{C}b$. We may assume $\\{x_{i}\\}_{i\in
I}=x^{\prime}\supseteq x$. Let
$\psi^{\prime}(x^{\prime},b)\in\mathrm{tp}(a/bC)$ Kim-fork over $C$. Then
$\tilde{\varphi}(x^{\prime},b)=:\psi(x,b)\vee\psi^{\prime}(x^{\prime},b)\in\mathrm{tp}(a/bC)$,
and $\tilde{\varphi}(x,b)$ Kim-forks over $C$. It suffices to show that
$\tilde{\varphi}(x^{\prime},b)$ has the property (*). Suppose
$\\{b_{\eta}\\}_{\eta\in\omega^{<\omega}}$, with $b_{\eta}\models p(y)$,
witness the failure of (*) for $\tilde{\varphi}(x^{\prime},b)$ and degree of
inconsistency $k$. By compactness, we can find
$\\{b_{\eta}\\}_{\eta\in\kappa^{<\omega}}$, where $\kappa=|b|^{+}$ (or
$\kappa=\omega$ if $b$ is finite) , so that for all
$\sigma\in\kappa^{\omega}$,
$\\{\tilde{\varphi}(x^{\prime},b_{\sigma|_{i}})\\}_{i<\omega}$ is consistent,
but for all $\eta\in\kappa^{<\omega}$,
$\\{\tilde{\varphi}(x^{\prime},b_{\eta\smallfrown\langle
i\rangle})\\}_{i<\kappa}$ is $k$-inconsistent. In particular, for all
$\eta\in\kappa^{<\omega}$ and all $j<m$,
$\\{\varphi_{j}(x,b_{\eta\smallfrown\langle i\rangle})\\}_{i<\kappa}$ is
$k$-inconsistent. We claim that for all $\eta\in\kappa^{\omega}$, there is
some $\gamma<\kappa$ so that, for all $j<m$,
$\varphi_{j}(x,b_{\eta\smallfrown\langle\gamma\rangle})$ is satisfied by no
tuple from $\\{b_{\eta|_{i}}\\}_{i<|\eta|}$. Suppose otherwise; because
$|\\{b_{\eta|_{i}}\\}_{i<|\eta|}|<\kappa$, by the pigeonhole principle there
is some fixed $j<m$ and tuple from $\\{b_{\eta|_{i}}\\}_{i<|\eta|}$ satisfying
$\varphi_{j}(x,b_{\eta\smallfrown\langle\gamma\rangle})$ for infinitely many
$\gamma$, contradicting $k$-inconsistency of
$\\{\varphi_{j}(x,b_{\eta\smallfrown\langle i\rangle})\\}_{i<\kappa}$. This
proves our claim; it follows by induction that there is some
$\sigma\in\kappa^{<\omega}$ so that for all $i<\omega$ and $j<m$,
$\varphi_{j}(x,b_{\sigma|_{i}})$ is satisfied by no tuple from
$\\{b_{\sigma|_{i^{\prime}}}\\}_{i^{\prime}<i}$. Extracting a
$C$-indiscernible sequence $\\{b_{i}\\}_{i<\omega}$ satisfying the same EM-
type over $C$ as $\\{b_{\sigma|_{i}}\\}_{i<\omega}$, we see that for all
$i<\omega$ and $j<m$, $b_{i}\models p(y)$ and $b_{<i}$ satisfies
$\varphi_{j}(x,b_{i})$–so $\\{b_{\sigma|_{i}}\\}_{i<\omega}$ 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 $C$ by choice of the
$\varphi_{j}(x,y)$–and $\\{\tilde{\varphi}(x^{\prime},b_{i})\\}_{i<\omega}$ is
consistent, though $\tilde{\varphi}(x^{\prime},b)$ Kim-forks over $C$ with
$b\models p(y)$, contradicting the direction (b) $\Rightarrow$ (a) of Theorem
7.9.

∎

Note that if we get rid of the assumption that $\mathrm{tp}(b/C)$ is isolated
or $T$ is low, the formula $\varphi(x,b)$ obtained in the above proof, though
not known to be simple as a formula in $\mathcal{L}(C)$, still does not have
the tree property given by a tree $\\{b_{\eta}\\}_{\eta\in\omega^{<\omega}}$
where $b_{\eta}\models p$.

In the setting of the $\mathrm{NSOP}_{1}$ variant of the Koponen conjecture,
Question 6.11, we now have get the following partial result as a corollary of
Theorem 7.20:

######  Corollary 7.21.

Let $T$ be $\mathrm{NSOP}_{1}$ with existence, and have quantifier elimination
in a finite relational language. Let $C$ be finite. Then for any finite tuple
$b$, there is some $n<\omega$ so that, for any $a=\\{a_{i}\\}_{i\in I}$ where
$|I|\geq n$ (with the $a_{i}$ allowed to repeat), and such 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{C}b$, there is some formula
$\varphi(x,b)\in\mathrm{tp}(a/Cb)$ that Kim-forks over $C$, where
$\varphi(x,y)$ is a formula with parameters in $C$ which is simple as a
formula of $\mathcal{L}(C)$.

This is because, by Lemma 6.5, any type over a finite set under the above
assumptions (which will of course be isolated) must have definable
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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.

Just as we defined types with algebraic forking in simple theories (Definition
2.1.7), we can ask whether there are types $p\in S(C)$ in an
$\mathrm{NSOP}_{1}$ theory with definable $\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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, but without algebraic Kim-forking–i.e. there are $a\models p$,
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{C}B$ so that
$\mathrm{acl}^{eq}(aC)\cap\mathrm{acl}^{eq}(BC)=\mathrm{acl}^{eq}(C)$, so not
all Kim-forking extensions of $p$ are given by the dependence relation coming
from $\mathrm{acl}^{eq}$. We give an example of such a $p$; this will be
isolated (and we suspect $T$ is low, which would mean it is not necessary that
$p$ is isolated), so we get a nontrivial instance of the simple Kim-forking
conjecture from Theorem 7.20

######  Example 7.22.

The following example is due to Conant and Kruckman ([21]). We show that it
gives us examples of an $\mathrm{NSOP}_{1}$ theory $T$ (with existence) so
that $1<\mathrm{max}(\\{F_{Mb}(p):p\in S(A),A\subset\mathbb{M}\\})<\infty$,
and with isolated types $p\in S(A)$ with $a\models p$ and $A\subset B$ so that
$B\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}^{K}a$ and
$\mathrm{bdd}(aA)\cap\mathrm{bdd}(B)=\mathrm{bdd}(A)$ (i.e., $p$ does not have
algebraic Kim-forking), and $p$ has definable Morley sequences. Since $p$ has
forking extensions where the forking is not witnessed by algebraic dependence,
but satisfies the hypotheses of theorem 7.20, this gives an instance of the
simple Kim-forking conjecture that is not covered by Kim’s result in [50] that
forking extensions where the forking is witnessed by algebraic dependence
satisfy the stable forking conjecture. Moreover, two other existing results on
the stable forking conjecture, those of Palacin and Wagner in [73] showing
that $\omega$-categorical CM-trivial supersimple theories satisfy the stable
forking conjecture, and of Peretz in [77] showing that the stable forking
conjecture holds between $\mathrm{SU}$-rank $2$ types in $\omega$-categorical
supersimple theories, both require $\omega$-categoricity to obtain a formula
$\varphi(x,y,z)$ so that for $a,b,c$ with $|a|=|x|$, $|b|=|y|$ and $|c|=|z|$,
$\models\varphi(a,b,c)$ if and only 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$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{c}b$. But for
finitely many variables $|x|$ and $a\models p$, it will follow from the
characterization of Kim-forking in this example that there is no formula
$\varphi(x,a,A)$ so that for $|b|=|x|$, $\models\varphi(b,a,A)$ if and only if
$b\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}^{K}a$. Finally, Brower in [16] obtains an
instance of the stable forking conjecture in simple theories eliminating
hyperimaginaries where $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}b$ with $\mathrm{SU}(a/C)<\omega$ and
$\mathrm{SU}(b/C)=2$, though the formula obtained is only stable in the sense
of $\mathcal{L}(C)$; to do so, he obtains a quasidesign between $a$ and $b$
(see Remark 7.23 below). Though in our example, Kim-forking is always given by
a quasidesign, the simple formula $\varphi(x,y)$ obtained in Theorem 7.20 does
not obviously define a quasidesign.

We recall the example of [21]. Let $\mathcal{L}$ be a language with two sorts
$P$ and $L$ and an incidence relation $I$ between singletons of $L$ and
singletons of $P$. For $m,n>1$, Conant and Kruckman show in [21] that the
theory of $\mathcal{L}$-structures omitting the complete bipartite graph
$K_{m,n}$ between $m$ points of $P$ and $n$ points of $L$ has a model
companion $T^{m,n}$. Call a $\mathcal{L}$ structure $A$ closed if for distinct
$p_{1},\ldots,p_{m}\in P(A)$ there are distinct $l_{1},\ldots,l_{n-1}\in L(A)$
so that $\models p_{i}Il_{j}$ for all $1\leq i\leq m$, $1\leq j\leq n-1$, and
for distinct $l_{1},\ldots,l_{n}\in L(A)$ there are distinct
$p_{1},\ldots,p_{m-1}\in P(A)$ so that $\models p_{i}Il_{j}$ for all $1\leq
i\leq m-1$, $1\leq j\leq n$. Then [21] show that the algebraically closed sets
in $\mathbb{M}$ are exactly the closed sets, and that $T$ is
$\mathrm{NSOP}_{1}$, with $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{A}b$ exactly when
$\mathrm{acl}(aA)\cap\mathrm{acl}(bA)=A$ and there are no
$a^{\prime}\in\mathrm{acl}(aA)\backslash A$,
$b^{\prime}\in\mathrm{acl}(bA)\backslash A$ so that $\models
a^{\prime}Ib^{\prime}$ or $b^{\prime}Ia^{\prime}$. Isomorphic algebraically
closed sets will satisfy the same type.

Assume without loss of generality that $m\leq n$. Suppose
$a\not\subseteq\mathrm{acl}(A)$, and let $p=\mathrm{tp}(a/A)$. We show that if
$\mathrm{acl}(aA)\backslash\mathrm{acl}(A)$ contains only points of $P$,
$F_{Mb}(p)=m$, and that if $\mathrm{acl}(aA)\backslash\mathrm{acl}(A)$
contains points of $L$, $F_{Mb}(p)=n$. Without loss of generality suppose we
are in the second case. We may also assume that $A\subseteq a$ and that $a$,
$A$ are algebraically closed. Fix a singleton $a_{0}\in L(a)\backslash A$. We
first show that $F_{Mb}(p)\geq n$. We can easily construct a $K_{m,n}$-free
$\mathcal{L}$-structure $B=\bigcup_{S\subset\omega,|S|\leq n-1}B_{S}$
satisfying the following:

(a) For all $S,T\subset\omega$ with $|S|,|T|\leq n-1$, $B_{S}\cap
B_{T}=B_{S\cap T}$, and for all $i<\omega$ and $b^{i}=B_{\\{i\\}}$, there are
$b^{i}_{0}\in b^{i}$ so that $b^{i}_{0}B^{i}B_{\emptyset}$ is isomorphic to
$a_{0}aA$;

(b) For all $S\subset\omega$ with $|S|\leq n-1$, $B_{S}$ is closed, and is a
minimal closed set containing $\cup_{i\in S}b^{i}$.

(c) For all $S\subset\omega$ with $|S|\leq n-1$ and $i\in\omega\backslash S$,
there are no singletons $b^{\prime}\in b^{i}\backslash B_{\emptyset}$ and
$b^{\prime\prime}\in B_{S}$ with $(b^{\prime},b^{\prime\prime})\in I(B)$ or
$(b^{\prime\prime},b^{\prime})\in I(B)$.

(Just choose freely so that (a) and (b) are satisfied; for example, if
$A=\emptyset$ and $a$ is a singleton of $L$, $B$ just consists of infinitely
many $L$-points.)

In $B$, there is no $b^{*}\in P(B)$ so that, for at least $n$ values of
$i<\omega$, $(b^{*},b^{i}_{0})\in I(B)$: such a $b^{*}$ must lie in some
$B_{S}$, but then for some $i\notin S$, $(b^{*},b^{i}_{0})\in I(B)$,
contradicting (c). So we may extend $B$ to a $K_{m,n}$-free
$\mathcal{L}$-structure $B\sqcup\\{b^{*}\\}$, so that $(b^{*},b)\in
I(B\cup\\{b^{*}\\})$ for $b\in B$ if and only if $b=b^{i}_{0}$ for some
$i<\omega$.

Now embed $B\sqcup\\{b^{*}\\}$ into $\mathbb{M}$ so that the image of
$B_{\emptyset}$ is $A$; let $a^{i}$ be the image of $b^{i}$, $a^{i}_{0}$ the
image of $b^{i}_{0}$, $a^{*}$ the image of $b^{*}$. Then for $0\leq
i_{1}<\ldots<i_{n+1}<\omega$, $a^{i_{j}}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{A}a^{i_{1}}\ldots a_{i_{j-1}}$ for $1\leq
j\leq n$, by (a)-(c). However, $a^{i_{n+1}}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{A}a^{i_{1}}\ldots a^{i_{n}}$, because
$a^{*}\in\mathrm{acl}(a^{i_{1}}\ldots a^{i_{n}})$ and $\models
I(a^{*},a_{0}^{i_{n+1}})$. Extracting an $A$-indiscernible sequence, we see
$F_{Mb}(p)\geq m$.

Now let us show that $F_{Mb}(p)\leq n$: let $I=\\{a_{i}\\}_{i<\omega}$ be an
$A$-indiscernible sequence of realizations of $p$, and suppose that $I$ is not
a Morley sequence over $A$. It suffices to show that
$a_{n}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{A}a_{<n}$. We may again assume that
$A\subseteq a$ and that $a$, $A$ are algebraically closed, and we may also
assume that $a_{i}\cap\mathrm{acl}(a_{<i})=A$ for all $i<\omega$, as otherwise
already $a_{1}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}^{K}a_{0}$ (see the proof of Proposition
2.1.10). Now for some $k<\omega$, $a_{k}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{A}a_{<k}$, so there is some
$a^{\prime}_{k}\in a_{k}\backslash A$ and $a^{\prime}\in
A^{\prime}=\mathrm{acl}(a_{<k})$ so that $\models
I(a^{\prime}_{k},a^{\prime})$, assuming without loss of generality that
$a^{\prime}\in L(A^{\prime})$. Then $\\{a_{i}\\}_{k\leq i<\omega}$ is
$A^{\prime}$-indiscernible, so for $0\leq i\leq n$ there is some
$a^{\prime}_{k+i}\in a_{k+i}\backslash A$ so that $\models
I(a^{\prime}_{k+i},a^{\prime})$. Then
$a^{\prime}\in\mathrm{acl}(a_{k},\ldots,a_{k+n-1})$, so because $\models
I(a^{\prime}_{k+n},a^{\prime})$, $a_{n+k}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{A}a_{k},\ldots,a_{k+n-1}$, implying
$a_{n}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{A}a_{<n}$ by $A$-indiscernibility. This
shows $F_{Mb}(p)\leq n$, so $F_{Mb}(p)=n$.

Now assume $\mathrm{acl}(aA)$ is finite, which implies that $p$ is isolated;
we observe that $p$ has definable Morley sequences; since $F_{Mb}(p)>1$, $p$
will also have forking extensions where the forking is not witnessed by
algebraic dependence. We may assume that $A,a$ are algebraically closed with
$A\subset a$; let $I=\\{a_{i}\\}$ be an indiscernible sequence of realizations
of $p$. Then by the above arguments and the characterization of Kim-forking,
$I$ is a Morley sequence if and only if $a_{0}\cap a_{1}=A$ and there are no
$a^{\prime}_{i}\in P(a_{i})$ for all $0\leq i\leq m$ together with some
$a^{*}$ so that $\models I(a^{\prime}_{i},a^{*})$ for all $0\leq i\leq m$, or
$a^{\prime}_{i}\in L(a_{i})$ for all $0\leq i\leq n$ together with some
$a^{*}$ so that $\models I(a^{*},a^{\prime}_{i})$ for all $0\leq i\leq n$.
This is definable, by finiteness of $a$.

######  Remark 7.23.

In Proposition 2.2.2 of [16], Brower identifies an instance of the stable
forking conjecture, based on the quasidesigns from [77], that generalizes the
case where the reason why $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}b$ is that
$\mathrm{acl}(aC)\cap\mathrm{acl}(bC)\neq\mathrm{acl}(C)$. Let $B(a,b,C)$
denote that

$a\notin\mathrm{acl}(bC)$, $b\notin\mathrm{acl}(aC)$ and for any nonconstant
$aC$-indiscernible sequence $I=\\{b_{i}\\}_{i<\omega}$ with $b_{0}=b$,
$a\in\mathrm{acl}(C)$.

In this case, by Proposition 2.2.2 of [16] there is a formula
$\varphi(x,b)\in\mathrm{tp}(a/Cb)$ which forks over $C$, so that
$\varphi(x,y)$ is stable in $\mathcal{L}(C)$.

In Example 7.22 above, any nontrivial instance of Kim-forking is always given
by an instance of $B(a,b,C)$. To express this, we define the following
property, which generalizes algebraic forking, Definition 2.1.7:

######  Definition 7.24.

Let $T$ be $\mathrm{NSOP}_{1}$ with existence. Then $p\in S(C)$ has the
_Brower property_ if whenever $a\models p$ 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{C}b$, either
$\mathrm{acl}^{eq}(aC)\cap\mathrm{acl}^{eq}(bC)\neq\mathrm{acl}^{eq}(C)$, or
there are some $a^{\prime}\in\mathrm{acl}^{eq}(aC)$ and
$b^{\prime}\in\mathrm{acl}^{eq}(bC)$ so that $B(a^{\prime},b^{\prime},C)$.

So the types $p$ with definable $\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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 discussed in Example
7.22, have the Brower property.

Note that, similarly to Proposition 2.1.9, that there are some
$a^{\prime}\in\mathrm{acl}(aC)$ and $b^{\prime}\in\mathrm{acl}(bC)$ so that
$B(a^{\prime},b^{\prime},C)$ implies 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{C}b$, so if $p$ has the Brower property,
whenever $a\models p$, $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{C}b$ if and only if either
$\mathrm{acl}(aC)\cap\mathrm{acl}(bC)\neq\mathrm{acl}(C)$, or there are some
$a^{\prime}\in\mathrm{acl}(aC)$ and $b^{\prime}\in\mathrm{acl}(bC)$ so that
$B(a^{\prime},b^{\prime},C)$. Moreover, by Proposition 2.2.2, Corollary 2.3.3
and Theorem 2.1.6 of Brower ([16]), if $p$ has the Brower property, $b\models
p$ 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{C}b$, then the conclusion of the stable
Kim-forking conjecture holds (the statements of Brower’s results are for
forking in simple theories, though the proofs will generalize): there is a
formula $\varphi(x,b)\in\mathrm{tp}(a/Cb)$ which Kim-forks over $C$, so that
$\varphi(x,y)$ is stable in $\mathcal{L}(C)$. We may then ask whether any
instance of the simple Kim-forking conjecture coming from Theorem 7.20
actually comes from the Brower property:

######  Question 7.25.

Let $T$ be $\mathrm{NSOP}_{1}$, and $p\in S(C)$ have definable
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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. Does $p$ have the
Brower property?

Note that in the context of Theorem 7.20, we may also ask this question under
the assumption that $T$ is low, or $p$ is isolated.

The examples of nontrivial instances of a type with definable
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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 seqeunces in Example 7.22 are
$\mathrm{NSOP}_{1}$, but not simple; we can ask whether simple or stable
theories have any nontrivial examples of this property. In a simple theory, we
adopt the convention of calling a type with definable
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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 a type with definable
Morley sequences.

######  Question 7.26.

Let $T$ be simple (or stable). Does every type $p\in S(A)$ with definable
Morley sequences have algebraic forking? Does every type $p\in S(A)$ with the
Brower property have algebraic forking?

We give a partial result on the first question. To motivate this, we extend
the degree of nonminimality to Kim-independence, by replacing every reference
to forking with a reference to Kim-forking.

######  Definition 7.27.

Let $T$ be $\mathrm{NSOP}_{1}$ with existence, and let $p\in S(A)$ be a type
admitting a nonalgebraic Kim-forking extension. The degree of nonminimality of
$p$ is the minimal length $n$ of a sequence of realizations of the type $p$,
say $a_{1},\ldots,a_{n}$ such that $p$ has a nonalgebraic Kim-forking
extension over $a_{1},\ldots,a_{n}$.

######  Example 7.28.

Let $T=K_{m,n}$, and let $p$ be the unique type over $\emptyset$ in, without
loss of generality, the point sort. In Example 7.22 we showed that $p$ (which
of course has nonalgebraic Kim-forking extensions) has definable
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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 and that $F_{Mb}(p)=m$;
we now show that $\mathrm{nmdeg}(p)=m$, so that $\mathrm{nmdeg}(p)=F_{Mb}(p)$.
Let $a_{1},\ldots a_{m}$ be distinct realizations of $p$; it suffices to show
that $a_{1}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}a_{2}\ldots a_{m}$. But
$a_{1}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}a_{2}\ldots a_{m}$ only if
$\mathrm{acl}(a_{1})=a_{1}$ has points incident to lines of
$\mathrm{acl}(a_{2}\ldots a_{m})=\\{a_{2}\ldots a_{m}\\}$, or vice versa,
impossible as both sets have only points, and no lines.

The above example motivates the following theorem:

######  Theorem 7.29.

Let $T$ be $\omega$-stable. Then there is no type $p\in S(A)$ such that
$\mathrm{RM}(p)$ is a limit ordinal, $\mathrm{nmdeg}(p)=F_{Mb}(p)$, and $p$
has definable Morley sequences.

###### Proof.

Because a set is independent over $A$ if and only if it is independent over
$\mathrm{acl}^{eq}(A)$, and is an indiscernible sequence over $A$ if and only
if it is an indiscernible sequence over $\mathrm{acl}^{eq}(A)$, the formula
witnessing that $p$ has definable Morley sequences also witnesses, for any
extension $p^{\prime}$ of $p$ to $\mathrm{acl}^{eq}(A)$, that $p^{\prime}$ has
definable Morley sequences, and moreover,
$\mathrm{nmdeg}(p^{\prime})=\mathrm{nmdeg}(p)=F_{Mb}(p)=F_{Mb}(p^{\prime})$
and $\mathrm{RM}(p)=\mathrm{RM}(p^{\prime})$ is a limit ordinal. So we may
assume $A$ is algebraically closed in $T^{eq}$, and therefore that $p$ is
stationary. If $p$ has definable Morley sequences, it follows that the set of
Morley sequences over $A$ consisting of realizations of $p$ is relatively
definable in the set of nonconstant $A$-indiscernible sequences consisting of
realizations of $p$. Let $\mathrm{nmdeg}(p)=F_{Mb}(p)=n$, and for
$q(y)=(p(x))^{\otimes{n}}$ the type of $n$ independent realizations of $p$
(which is well-defined by stationarity of $p$), let $\sigma$ be the set of
formulas $\varphi(x,y)$ where $\varphi(x,a)$ forks over $A$ for any $a\models
q(y)$. Since $\mathrm{nmdeg}(p)=F_{Mb}(p)=n$, a nonconstant $A$-indiscernible
sequence $\\{a_{i}\\}_{i<\omega}$ is a Morley sequence if and only if
$a_{n}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{0}\ldots a_{n-1}$, if and only if
$\models\neg\varphi(a_{n},a_{0}\ldots a_{n-1})$ for every
$\varphi(x,y)\in\sigma$. By compactness, and the assumption that $p$ has
definable Morley sequences over $a$, there is then some $\varphi(x,y)$ so
that, for any $a_{0}\ldots a_{n-1}\models q(y)$, $\varphi(x,a_{0}\ldots
a_{n})$ forks over $A$, and so that, for $\\{a_{i}\\}_{i<\omega}$ a
nonconstant $A$-indiscernible sequence, $\\{a_{i}\\}_{i<\omega}$ is not an
$A$-Morley sequence if and only if $\models\varphi(a_{n},a_{0}\ldots
a_{n-1})$. Fix some $b_{0}\ldots b_{n}\models q(y)$, and let
$\mathrm{RM}(p)=\lambda$; we have assumed that $\lambda$ is a limit ordinal.
Then $\varphi(x,b_{0}\ldots b_{n})$ forks over $A$, so
$\lambda^{\prime}=:\mathrm{RM}(\varphi(x,b_{0}\ldots b_{n})\cup
p(x))<\lambda$, because otherwise there would be a completion $\tilde{p}(x)\in
S(Ab)$ of $\varphi(x,b_{0}\ldots b_{n})\cup p(x)$ so that
$\mathrm{RM}(\tilde{p(x)})=\mathrm{RM}(\varphi(x,b_{0}\ldots b_{n})\cup
p(x))=\mathrm{RM}(p(x))$, implying that $\tilde{p}(x)$ does not fork over $A$
despite containg $\varphi(x,b_{0}\ldots b_{n})$, a contradiction. Since
$\lambda$ is a limit ordinal, there is an ordinal $\lambda^{\prime\prime}$ so
that $\lambda^{\prime}<\lambda^{\prime\prime}<\lambda$. There is then some
$q(x)\in S(B)$ extending $p(x)$ so that
$\mathrm{RM}(q(x))=\lambda^{\prime\prime}$. Let $\\{a_{i}\\}_{i<\omega}$ be a
Morley sequence in $q(x)$ over $B$. Then $\\{a_{i}\\}_{i<\omega}$ is
nonconstant, but is not a Morley sequence over $A$, because if it were, by
Kim’s lemma $B\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{A}a_{0}$, contradicting that $a_{0}\models
q(x)$, which forks over $A$. However, $\mathrm{RM}(a_{n}/a_{0}\ldots
a_{n-1})\geq\mathrm{RM}(a_{n}/Ba_{0}\ldots
a_{n-1})=\mathrm{RM}(a_{n}/B)=\mathrm{RM}(q(x))=\lambda^{\prime\prime}$, the
first equality because $\\{a_{i}\\}_{i<\omega}$ is a Morley sequence over $B$.
However, because $\\{a_{i}\\}_{i<\omega}$ is nonconstant and not a Morley
sequence over $A$, $\varphi(x,a_{0}\dots
a_{n-1})\in\mathrm{tp}(a_{n}/a_{0}\ldots a_{n-1})$, so
$\mathrm{RM}(\varphi(x,a_{0}\ldots a_{n})\cup
p(x))\geq\mathrm{RM}(a_{n}/a_{0}\ldots a_{n-1})\geq\lambda^{\prime\prime}$.
But $a_{0}\ldots a_{n-1}\equiv_{A}b_{0}\ldots b_{n-1}$, so
$\lambda^{\prime}=\mathrm{RM}(\varphi(x,b_{0}\ldots b_{n})\cup
p(x))=\mathrm{RM}(\varphi(x,a_{0}\ldots a_{n})\cup
p(x))\geq\lambda^{\prime\prime}>\lambda^{\prime}$, a contradiction.

∎

######  Example 7.30.

Since it is open whether there are $\omega$-stable theories with types $p$ so
that $1<F_{Mb}(p)<\infty$, it is unknown whether the hypotheses of this
theorem can be satisfied with $\mathrm{nmdeg}(p)>1$. However, one special case
of $\mathrm{nmdeg}(p)=F_{Mb}(p)$ is where $F_{Mb}(p)=1$. In example 4.11, the
unique type $p\in S(\emptyset)$, the $\omega$-stable free pseudoplane from
example 7.2.10 of [77], is exhibited as a type $p$ without algebraic forking
and with $F_{Mb}(p)=1$; moreover, it has $\mathrm{RM}(p)=\omega$, a limit
ordinal. So we have found a type (without algebraic forking) whose failure to
have definable Morley sequences is a consequence of Theorem 7.29.

We now consider the relationship between definable $\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\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 and type-counting. In $\omega$-categorical theories, the following
converse to Remark 7.19 is immediate:

######  Proposition 7.31.

Let $T$ be an $\omega$-categorical $\mathrm{NSOP}_{1}$ theory with existence,
and let $A$ be finite. Then $p\in S(A)$ has definable
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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 if and only if
$F_{Mb}(p)<\omega$.

This leads us to ask if there are any nontrivial examples of this equivalence,
or more generally:

######  Question 7.32.

Let $T$ be an $\omega$-categorical $\mathrm{NSOP}_{1}$, and let
$F_{Mb}(p)<\omega$. Does $p$ have algebraic forking (and therefore,
$F_{Mb}(p)=1$)?

However, we do not need all of $\omega$-categoricity for the equivalence in
Proposition 7.33, only a weaker claim about type-counting. The following
stronger statement follows easily from the definition of definable
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\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:

######  Proposition 7.33.

Let $T$ be an $\mathrm{NSOP}_{1}$ theory with existence, and let $A$ be
finite. If $p\in S(A)$, and $F_{Mb}(p)$ and $|S^{F_{Mb}(p)+1}(A)|$ are finite,
then $p$ has definable Morley sequences.

Unlike the $\omega$-categorical case, we know we have nontrivial examples of
this:

######  Example 7.34.

Let $T=K_{m,2}$, and let $p$ be the unique type over $\emptyset$ in the point
sort. Then $F_{Mb}(p)=m$, and we show that $|S^{F_{Mb}(p)+1}(\emptyset)|$ is
finite, where $S^{F_{Mb}(p)+1}(\emptyset)$ is taken in the point sort. Let
$p_{1},\ldots,p_{k}$ be points where $k\leq m+1$, and let
$l_{1},\ldots,l_{\ell}\in\mathrm{acl}(p_{1}\ldots p_{m+1})$ be the at most
$m+1$ many lines obtained by finding a line incident to any $m$ points in
$p_{1},\ldots,p_{m+1}$. Since isomorphic algebraically closed sets satisfy the
same type, it suffices to show that $\\{p_{1},\ldots,p_{k},l_{1}\ldots
l_{\ell}\\}$ is algebraically closed. This will hold as long as two criteria
are satisfied: first, any $m$ of the $p_{1},\ldots,p_{k}$ are incident to a
common line of $l_{1}\ldots l_{\ell}$, which follows by construction, and
second, any two of the $l_{1}\ldots l_{\ell}$ are incident to $m-1$ of the
$p_{1},\ldots,p_{k}$. But any individual line in the $l_{1}\ldots l_{\ell}$
is, by construction, incident to $m$ of the $p_{1},\ldots,p_{k}$, so because
$k\leq m+1$, any two lines of the $l_{1}\ldots l_{\ell}$ must be incident to
$m-1$ of the $p_{1},\ldots,p_{k}$.

The nontriviality of this example motivates the following corollary of Theorem
7.20 (note that isolatedness of $p$ follows from the hypotheses):

######  Corollary 7.35.

Let $T$ be $\mathrm{NSOP}_{1}$ with existence, and let $A$ be finite. Let
$p\in S(C)$, and let $F_{Mb}(p)$ and $|S^{F_{Mb}(p)+1}(C)|$ be finite. Then
there is some $n<\omega$ so that, for any $a=\\{a_{i}\\}_{i\in I}$ where
$|I|\geq n$ (with the $a_{i}$ allowed to repeat), and such 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$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{C}b$, there is some formula
$\varphi(x,b)\in\mathrm{tp}(a/Cb)$ that Kim-forks over $C$, where
$\varphi(x,y)$ is a formula with parameters in $C$ which is simple as a
formula of $L(C)$.

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