in section 5 we will discuss some interesting examples of strictly
$\mathrm{NSOP}_{4}$ theories not covered by these cases.

Note that a related notion called “Conant-independence” is defined using
finitely satisfiable Morley sequences in [21], where it is shown to coincide
with Kim-independence in $\mathrm{NSOP}_{2}$ theories. Despite the fact that
the choice between invariant or finitely satisfiable Morley sequences does not
matter for Kim-independence in $\mathrm{NSOP}_{1}$ theories (see Fact 2.1), it
is not known when our notion of Conant-independence and the one from [21]
coincide222We suspect that they do not, even in $\mathrm{NSOP}_{4}$ theories.
If Conant-independence with respect to finitely satisfiable Morley sequences
coincided with the standard Conant-independence
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$ in the
triangle-free random graph, and there were also minimal finitely satisfiable
types in restriction of the Kim-dividing order to finitely satisfiable types,
then $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$ would have
to satisfy a “weak independence theorem” (see Proposition 6.10 of [15] for the
original result, or proposition 5.7 of [21], whose proof is quoted below, for
a result involving Conant-independence with respect to finitely satisfiable
Morley sequences) with respect to those minimal finitely satisfiable types.
But satisfying a weak independence theorem for
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$ can be seen
in this example to characterize free amalgamation (the standard one, with no
new edges). We do not think that the invariant types given by free
amalgamation in the triangle-free random graph are finitely satisfiable..

######  Example 3.14.

In an $\mathrm{NSOP}_{4}$ theory, or even a theory with symmetric Conant-
independence, the relation $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ might satisfy
the generalized freedom axiom but not the equivalent conditions of Theorem
3.6. Consider the theory of the generic $K_{3}$-free graph with two constants,
$c$ and $d$, for distinct vertices. This is $\mathrm{NSOP}_{4}$, originally by
work of Shelah ([27]). Declare $A\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}B$ if $A\cap
B\subseteq M$ and a node $a$ of $A\backslash M$ and a node $b$ of $B\backslash
M$ have an edge between them if and only if $a$ connects to $c$ (denoted by
its constant), $b$ connects to $d$ (denoted by its constant) and $a$ and $b$
connect to no common vertices in $M$. This makes sense as a stationary
relation, as the fact that $a$ and $b$ connect to no common vertices in $M$
tells us that no triangles can be formed with vertices in $M$, and the fact
that two vertices of $A\backslash M$ or $B\backslash M$ must connect to a
common constant, so must not be connected themselves, if they are to recieve
any new edges, tells us that no triangles can be formed without vertices in
$M$. We will first show the generalized freedom axiom. Suppose there is an
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Morley sequence
$\\{A_{i}\\}_{i\in\omega}$ of copies of a set $A$ over $M$ realizing the type
of $A$ over $M^{\prime}\succ M$. Then clearly $A_{i}\cap M^{\prime}\subseteq
M$ and $A_{i}\cap A_{j}\subseteq M$ for $i\neq j$, and it remains to show that
the edges between nodes $a$ of $A_{i}\backslash M$ and $b$ of $A_{j}\backslash
M$ remain as selected according to $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$, when considered
over $M^{\prime}$. If there is no edge between $a$ and $b$, then the
conditions for $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ to dictate this
remain clearly remain true over $M^{\prime}$. On the other hand, if $a$ and
$b$ are related by an edge, then (assuming $i>j$) $a$ must be connected to $c$
and $b$ to $d$, but they must still not be connected to any common nodes even
in $M^{\prime}\backslash M$, because then a triangle would be formed. So the
conditions for an edge according to $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ remain true over
$M^{\prime}$ in this case, too.

However, $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ does not satisfy
the relative Kim’s lemma (so neither is relative Kim-independence symmetric).
Consider distinct disconnected vertices $\overline{c}=\\{c_{1},c_{2}\\}$
outside of $M$, $c_{1}$ connected only to $c$ and to no other vertices of $M$,
$c_{2}$ connected only to $d$ and to no other vertices of $M$. Consider the
formula $\varphi(x,\overline{c})=:x\neq c_{1}\wedge x\neq c_{2}\wedge
xRc_{1}\wedge xRc_{2}$. Clearly this does not Kim-dividide with respect to the
standard free amalgamation given by adding no new edges. But it
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Kim-divides, as
if $\overline{c}^{1},\overline{c}^{2}$ begin an
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Morley sequence
of copeies of $\overline{c}$ over $M$, then $c^{2}_{1}$ and $c^{1}_{2}$ are
related by an edge, making it impossible for some other vertex to connect to
both of them.

Therefore, we cannot get a witnessing lemma for stationary independence
relations for $\mathrm{NSOP}_{4}$ theories of the kind we obtained for
$\mathrm{NSOP}_{2}$ theories in Theorem 4.3 of [21] in order to prove the
$\mathrm{NSOP}_{1}$-$\mathrm{SOP}_{3}$ dichotomy (so in that case,
$\mathrm{NSOP}_{1}$). Moreover, this example tells us that even though the
equivalent conditions of Theorem 3.6 imply that
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Kim independence
is just Conant-independence, in the statement of Theorem 3.6 we must still
consider $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Kim-independence
itself and not just Conant-independence, as Conant-independence can be
symmetric (perhaps due to the presence of a relation
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{1}$ with the
generalized freedom axiom and the relative Kim’s lemma, relative Kim-
independence with respect to which will then be Conant-independence; see
Example 3.2 for the triangle-free random graph), even if there is some
relation $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{2}$ (which will
then be different from $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{1}$, such as in
the relation constructed in this example) with the generalized freedom axiom
but without the relative Kim’s lemma, so that
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{2}$-Kim
independence, which will not be Conant-independence, remains asymmetric.

We conclude this section by isolating two model-theoretic assumptions, related
by the generalized freedom axiom as in Theorem 3.6 and without any known
$\mathrm{NSOP}_{4}$ counterexamples, which together imply that a theory must
be either $\mathrm{NSOP}_{1}$ or $\mathrm{SOP}_{3}$, and either
$\mathrm{TP}_{2}$ or simple.

######  Definition 3.5.

We say a theory $T$ has the strong witnessing property if for
$M\prec\mathbb{M}$ there is some $\mathbb{M}_{1}\succ M$ so that for
$b\subset\mathbb{M}_{1}$, $\mathrm{tp}(b/\mathbb{M})$ is an $M$-invariant
extension of $\mathrm{tp}(b/M)$ minimal in the Kim-dividing order.

Note that if $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{i}$ has left
extension, then whether a formula $\varphi(x,b)$ Kim-divides over $M$ is
unchanged under adding or removing unused parameters, so under this assumption
the strong witnessing property is satisfied as long as every type over $M$ has
an $M$-invariant extension minimal in the Kim-dividing order.

######  Theorem 3.15.

If a theory $T$ satisfies the strong witnessing property and has symmetric
Conant-independence, then it is either $\mathrm{NSOP}_{1}$ or
$\mathrm{SOP}_{3}$.

###### Proof.

If $p(x)$ is a type over $M$, then define a strong witnessing extension of
$p(x)$ to be a global extension $q(x)$ of $p(x)$ so that, for all tuples
$b\in\mathbb{M}$ if $c\in\mathbb{M}$ with $c\models q(x)|_{Mb}$, then for any
$a\in\mathbb{M}$ there is $a^{\prime}\equiv_{Mc}a$ with
$a^{\prime}\in\mathbb{M}$ so that $\mathrm{tp}(a^{\prime}c/Mb)$ extends to an
$M$-invariant type minimal in the Kim-dividing order among $M$-invariant
extensions of $\mathrm{tp}(a^{\prime}c/M)=\mathrm{tp}(ac/M)$. By the strong
witnessing property, strong witnessing types extending any $p(x)$ exist (see
Lemma 5.5 of [21]). Conant-dividing is the same as Kim-dividing witnessed by a
Morley sequence in some (any) strong witnessing type, and Conant-forking is
the same as Conant-dividing as in Proposition 3.5. Meanwhile, Conant-
independence is symmetric by assumption, and the chain condition for Conant-
independence with respect to Morley sequences in strong witnessing types is as
in Claim 3.7. So the result follows nearly word-for-word from the proofs of
Proposition 5.7 of [21] and the discussion in Section 6 in the same paper,
just replacing any reference to the coheir notions of Conant-independence and
Kim-dividing independence with the invariant notions, and replacing any
reference to canonical coheirs and canonical Morley sequences with strong
witnessing types and invariant Morley sequences in those types.

Note that the proof of Proposition 5.7 of [21] comes directly from the proof
of the “weak independence theorem” (Proposition 6.10) of Kaplan and Ramsey in
[15]. It plays the role in our argument that the freedom axiom plays in
Theorem 7.17 of [5], showing that a modular free amalgamation theory must be
either simple or $\mathrm{SOP}_{3}$. The proof of that theorem serves as a
basis for section 6 of [21], which requires an entirely new argument in Claim
6.1 of that section. ∎

The following follows Theorem 7.7 of Conant ([5]), using a similar argument:

######  Theorem 3.16.

If a theory $T$ satisfies the strong witnessing property and has symmetric
Conant-independence, then it is either $\mathrm{TP}_{2}$ or simple.

###### Proof.

If $T$ is not simple then dividing-independence is not symmetric, so because
Conant-independence is symmetric, there are
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{K^{*}}_{M}b$ and
indiscernible sequence $I=\\{b_{i}\\}_{i<\kappa}$ for very large $\kappa_{1}$,
starting with $b$, and $\varphi(x,b)\in\mathrm{tp}(a/Mb)$ so that
$\\{\varphi(x,b_{i})\\}_{i\in I}$ is $k$-inconsistent for some $k$. Find a
copy pver $I$, $I^{\prime}\equiv_{M}I$, in $\mathbb{M}_{1}$ from the
definition of the strong witnessing property, and let
$q(\overline{y})=\mathrm{tp}(I^{\prime}/\mathbb{M}_{1})$. By the pigenonhole
principle we can assume that each term of $I$ satisfies the same type
$q_{1}(y)$ over $\mathbb{M}$. Now take an invariant Morley sequence in
$q(\overline{y})$ over $M$. Taking the each copy of $I$ as a row, which will
then give us an inconsistent set of instances of $\varphi(x,y)$, we see also
that the paths are consistent, being invariant Morley sequences in the type
$q(y)$ which is minimal in the Kim-dividing order, with $\varphi(x,b)$ not
Conant-dividing over $M$. ∎

######  Corollary 3.16.1.

Suppose $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ satisfies the
generalized freedom axiom and the relative Kim’s lemma. Then $T$ is either
$\mathrm{NSOP}_{1}$ or $\mathrm{SOP}_{3}$, and is either simple or
$\mathrm{TP}_{2}$.

######  Remark 3.17.

Under the hypotheses of the corollary, applying the generalized freedom axiom
rather than the chain condition in the proof of the “weak independence
theorem” analogue, we get (by stationarity) a “base monotone” version of this
result (see section 2 of [20] for related results on “base monotone” versions
of independence): if $M^{\prime}\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{K^{*}}_{M}a$ and
$M^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{K^{*}}_{M}b$ and
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M^{\prime}}b$,
then $a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}b$. When in
addition $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{K^{*}}=\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$, note the
resemblance to the case of the freedom axiom where $C=M^{\prime}$, $D=M$ are
models and $C\cap AB\subseteq C\cap\mathrm{acl}(AD)\mathrm{acl}(BD)=D\subseteq
C$.

##  4\. Non-modular free amalgamation theories

The following property of relations $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ between sets can
be found in the “full transitivity” from section 2:

######  Definition 4.1.

The relation $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ has base
monotonicity if $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{B}C$ and
$B\subseteq D\subseteq C$ then $A\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}C$.

This is Proposition 8.8 of [15]:

######  Fact 4.1.

An $\mathrm{NSOP}_{1}$ theory is simple if and only if Kim-independence
satisfies base monotonicity for $B=M\prec M^{\prime}=D$ models.

Conant asks ([5], Question 7.19) if any free amalgamation theory is modular:

######  Definition 4.2.

A theory is _modular_ if $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$ has base
monotonicity.

Answering this question, we give an example of a nonmodular free amalgamation
theory. Kruckman and Ramsey ([20]) show that the empty theory in a language
with a binary function symbol $f(x,y)$ has a strictly $\mathrm{NSOP}_{1}$
model completion, where Kim-independence coincides with
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$ and the
algebraic closure coincides with closure under $f(x,y)$. This theory is
therefore non-modular. To show that forking coincides with dividing for
complete types over models, they introduce a relation of “free amalgamation”
that we expect to satisfy the generalized freedom axiom (see example 3.4), but
it is not a free amalgamation relation in the sense of [5]. We define a
nonstandard relation $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$, which will
satisfy the free amalgamation axioms. Let $T$ be the model completion of the
empty theory in the language with a binary function symbol, with an additional
constant symbol $c$. Define $A\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}B$ to mean
$A\cap B\subseteq C$ and, for $a\in A\backslash C$ and $b\in B\backslash C$,
$f(b,a)=f(a,b)=c$. We show that $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ is a free
amalgamation relation. Invariance through full transitivity are
straightfoward. For full existence, we can enlarge $a,B$ to their algebraic
closure with $C$, which we assume is algebraically closed. We can easily find
a structure in the language extending $C$ where $a$ and $B$ embed disjointly
over $C$, and where a point with coordinates properly in each of $a$ and $B$
will have image $c$. Full existence then follows from the fact that $T$ is the
model completion. If $C\subseteq a\cap b$, $a,b,C$ algebraically closed, then
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}b$ determines
the isomorphism type of $ab$ over $C$, so stationarity follows from quantifier
elimination. For freedom, if $A\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}B$ and $C\cap
AB\subseteq D\subseteq C$, then $A\cap B\subseteq C\cap AB\subseteq D$, while
for $a\in A\backslash D\subseteq A\backslash C$, $b\in B\backslash D\subseteq
B\backslash C$, $f(a,b)=f(b,a)=c$ as before. Finally, if
$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}B$ for
$C\subseteq A\cap B$ and $A,B,C$ algebraically closed, then the closure under
$f(x,y)$, and therefore the algebraic closure, of $AB$ remains $AB$, yielding
the closure axiom.

The existence of non-modular free amalgamation theories motivates the
following generalization of Theorem 7.17 of [5] that modular free amalgamation
theories are either simple or $\mathrm{SOP}_{3}$:

######  Theorem 4.2.

Free amalgamation theories are either $\mathrm{NSOP}_{1}$ or
$\mathrm{SOP}_{3}$.

First, we observe that, justifying the terminology, Lemma 7.6 of Conant in [5]
is essentially a characterization of Conant-independence:

######  Proposition 4.3.

Conant-independence in free amalgamation theories coincides with
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$ over models.

###### Proof.

Clearly Conant-independence implies $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$. Conversely,
suppose $a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}_{M}b$. By
extension for $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$, we can
assume that $a$ and $b$ are algebraically closed sets containing $M$. But then
by Lemma 7.6 of [5], there will be an $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Morley sequence
over $M$ (extending Defintion 3.1 appropriately) starting with $b$ and
indiscernible over $Ma$, so $a\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{K^{*}}_{C}b$. ∎

We also see that free amalgamation theories satisfy the strong witnessing
property (Definition 3.5): a formula $\varphi(x,b)$ not Conant-dividing over
$M$ must have a realization $a$ with $a\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}_{M}b$, and
then we can proceed as in the above proof. So the theorem follows from Theorem
3.15.

If a free amalgamation theory $T$ is $\mathrm{NSOP}_{1}$, then by Fact 2.1 and
Proposition 4.3, $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{K}=\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{K^{*}}=\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$. Thus the
characterization of simple theories within the class $\mathrm{NSOP}_{1}$ in
Fact 4.1 gives us the following, extending Conant’s result [5] that a simple
free amalgamation theory is modular:

######  Proposition 4.4.

An $\mathrm{NSOP}_{1}$ free amalgamation theory is modular if and only if it
is simple.

##  5\. Some examples

We consider two examples of theories with relations satisfying the assumptions
at the beginning of Section 3, as well as the generalized freedom axiom and
the relative Kim’s lemma, and characterize Conant-independence in these
structures. Our purposes are twofold: to give a model-theoretic interpretation
of certain tame independence relations in potentially strictly
$\mathrm{NSOP}_{4}$ theories, and to extend the concept of free amalgamation
to examples not covered by Conant’s work in [5].

######  Example 5.1.

(Countably categorical Hrushovski constructions.) We consider the case of the
examples of $\omega$-categorical structure with a predimension introduced in
section 3 of [12], which is developed in [13]. Let $\mathcal{L}$ be a language
with finitely many relations ([13] only require finitely many relations of
each arity, but we include this requirement so that the predimension function
only takes a discrete set of values), and for each relation symbol $R_{i}$,
let $\alpha_{i}$ be a non-negative real number associated to $R_{i}$. 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 $A$, and define the relation $A\leq B$ for $A$ a
finite $\mathcal{L}$-structure and $B$ any $\mathcal{L}$-structure 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 any
substructure $A$ of which satifies $d_{0}(A)\geq f(A)$. Assume that, if
$B_{1}\geq A\leq B_{2}$ belong to $\mathcal{C}_{f}$, then their evident “free
amalgamation,” by taking their disjoint union over $A$ and adding no new
edges, likewise belongs to $\mathcal{C}_{f}$. Then there is a
$\mathcal{L}$-structure $M$ every finite substructure of which belongs to
$\mathcal{C}_{f}$ and so that if $B\geq A\leq M$ with $B$ finite, then there
is an embedding $\iota:B\to M$ over $A$ so that $\iota(B)\leq M$. Let $T$ be
its (complete) theory. The theory $T$ is $\omega$-categorical, so has bounded
algebraic closure, and isomorphic algebraically closed sets are elementarily
equivalent. For $M$ a model of $T$, and $A\subseteq B\subseteq M$ with $A$
finite and $B$ any set, $A$ is algebraically closed in $B$ if $A\leq B$, and
$M$ will always continue to have the property that if $B\geq A\leq M$ with $B$
finite, then there is an embedding $\iota:B\to M$ over $A$ so that
$\iota(B)\leq M$.

Though $T$ is not necessarily simple, [13] show that it is either strictly
$\mathrm{NSOP}_{4}$ or simple. However, it does have a natural notion of
independence, even in the strictly $\mathrm{NSOP}_{4}$ case. We first recall
an additional property of an abstract relation
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ between sets:

######  Definition 5.1.

We say $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ has _finite
character_ if $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{B}C$ holds
whenever $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{B}C_{0}$ holds
for all finite $C_{0}\subseteq C$.

This notion of independence, called $d$-independence, is defined in [12]; it
will coincide with forking-independence in the simple case. For finite $A,B$,
denote $d(A/B)=d_{0}(\mathrm{acl}(AB))-d_{0}(\mathrm{acl}(B))$ (recalling the
bounded algebraic closure). This notion of relative dimension has a natural
extension over infinite sets: for $A$ a finite set and $B$ any set, denote
$d(A/B)=\min(\\{d(A/B_{0}):B_{0}\subseteq B\>\mathrm{finite}\\})$. We use the
following notation for the relation referred to in [12] as $d$-independence:
for $a$ finite and $B,C$ any sets, let
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}_{B}C$ if and
only if $d(a/BC)=d(a/B)$ and
$\mathrm{acl}(aB)\cap\mathrm{acl}(CB)=\mathrm{acl}(B)$; for $a$, $B$, $C$
finite this last condition will be redundant. In [12] it is shown that this
has finite character and is symmetric, monotone and fully transitive where
defined, so it extends naturally to a relation defined for $a$ possibly
infinite with the same properties. We claim that there is a natural relation
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ satisfying the
assumptions at the beginning of section 3 as well as the generalized freedom
axiom and the relative Kim’s lemma, and that Conant-independence coincides
with $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}$ (so is in
particular, symmetric).

We first observe a variant of property (P5) of [12] which, in place of a
finitary analogue of the “independence theorem” holding only in the simple
examples, constitutes a base-monotone version of the “weak independence
theorem” with respect to free amalgamation analogous to those in [15] with
respect to coheir-independence or [21] with respect to canonical coheirs. This
is used implicitly in [13] to show $\mathrm{NSOP}_{4}$, but we provide some
justification.

(P5’) Let $B_{1}\geq A\leq B_{2}$ be finite algebraically closed sets such
that $B_{1}$ and $B_{2}$ are freely amalgamated over $A$, which is to say
$\mathrm{acl}(B_{1}B_{2})$ is the disjoint union of $B_{1}$ and $B_{2}$ over
$A$ with no new relations. Let $c_{1}$, $c_{2}$ be finite with
$c_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}_{A}B_{1}$,
$c_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}_{A}B_{2}$
with $c_{1}\equiv_{A}c_{2}$; then there is some $c$ realizing
$\mathrm{tp}(c_{1}/B_{1})\cup\mathrm{tp}(c_{2}/B_{2})$–with
$c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{d}_{A}B_{1}B_{2}$ (which is not needed here)–and
$\mathrm{acl}(cB_{1})$ and $\mathrm{acl}(cB_{2})$ freely amalgamated over
$\mathrm{acl}(cA)$.

When we only require that $B_{1}\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}B_{2}$ rather
than that they be freely amalgamated, this is shown in Theorem 3.6(ii) of [12]
under assumptions on $f$, so we need only observe that this proof works for
this partial result without the assumptions on $f$. As in that proof we can
form the $\mathcal{L}$-structure $F=E_{12}\cup E_{13}\cup E_{23}$ with no new
relations, and with compatible isomorphisms
$\varphi_{12}:\mathrm{acl}(B_{1}B_{2})\to E_{12}$,
$\varphi_{j3}:\mathrm{acl}(c_{j}B_{j})\to E_{j3}$, which will be a special
case of the construction from that proof where the “underlying” predimension
$y$ is just the cardinality. Now by point (i) of that proof, which does not
use the additional assumption on $f$ required for simplicity, $E_{ij}\leq E$.
The part of the proof where this additional assumption is required is point
(ii), where it is shown that $F\in\mathcal{C}_{f}$; it must be shown that for
each $D\subseteq F$, $d_{0}(D)\geq f(|D|)$. However, the assumption on $f$ is
only used when $D$ is not contained in the union of two of the $E_{ij}$ (where
the requirement follows by closure under free amalgamation). But $F=E_{13}\cup
E_{23}$ because $\mathrm{acl}(B_{1}B_{2})=B_{1}\cup B_{2}$. So embedding a
copy of $F$ over $B_{1}B_{2}$ (where $B_{1}B_{2}$ is identified by its image
in $E_{12}\subset F$) so that it is algebraically closed will realize both
types, and in a $d$-independent way by point (iii), which does not rely on the
additional assumptions on $f$.

Now note that for $B_{1}\geq A\leq B_{2}$ algebraically closed finite sets and
$c$ any finite set with $\mathrm{acl}(cB_{1})$ and $\mathrm{acl}(cB_{2})$
freely amalgamated over $\mathrm{acl}(cA)$, the type of $cB_{1}$ and $cB_{2}$
then completely determine the type of $cB_{1}B_{2}$ and in particular
$B_{1}B_{2}$, so (P5’) implies that $B_{1}$ and $B_{2}$ are freely amalgamated
over $A$. This observation leads to the following definition: for $M$ a model
and $b,c$ finite sets of parameters, say
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}b$ if for any
finite $A\leq M$ with $a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}_{A}M$,
$b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}_{A}M$ (such
an $A$ always exists because $d_{0}$ only takes a discrete set of values; see
Lemma 2.17(a)(ii) of [12]), $\mathrm{acl}(aA)$ and $\mathrm{acl}(bA)$ are
freely amalgamated over $a$. For existence, by compactness, it suffices to
show that for types $p(x)$ and $q(y)$ over $M$, finitely many finite
$A_{i}\leq M$ such that $p(x)$ and $q(x)$ $d$-independently extend their
restrictions to $A_{i}$, and finite $B\subseteq M$, there are realizations $a$
of $p(x)|_{B}$ and $b$ of $q(y)|_{B}$ so that $\mathrm{acl}(aA_{i})$ and
$\mathrm{acl}(bA_{i})$ are freely amalgamated over $A_{i}$ for each $i$. But
take any $A\leq M$ containing each of the $A_{i}$ and $B$ and take
realizations $a$ of $p(x)|_{A}$ and $b$ of $q(y)|_{A}$ so that
$\mathrm{acl}(aA)$ and $\mathrm{acl}(bA)$ are freely amalgamated over $A$;
then the free amalgamation conditions over the $A_{i}$, by the observation at
the beginning of this paragraph, will be satisfied. Since by the quantifier
elimination, this relation is clearly stationary, and it is monotone by the
properties of free amalgamation for finite sets, it extends to a relation
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}b$ for $a$,
$b$ potentially infinite.

We next show that if $M^{\prime}\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}_{M}a$ and
$\\{a_{i}\\}_{i\in I}$ is an $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Morley sequence
starting with $a$ over $M^{\prime}$, then it is an
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Morley sequence
over $M$. But this follows from the definitions, by the fact that if $A\leq M$
is finite with $a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}_{A}M$ and
$M^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}_{M}a$, then
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{d}_{A}M^{\prime}$ (transitivity and symmetry). So we have the
generalized freedom axiom. We can also carry out a similar proof for a set in
place of $M^{\prime}$ (which we can assume to be algebraically closed and
contain $M$), so $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}$ implies
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Kim
independence.

We next show that $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Kim independence
implies $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}$: if
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}$ fails, the
proof from [12] that this implies dividing-independence (Lemma 2.19 (a))
relies on [18], which will tell us than any
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}$-independent
sequence witnesses dividing. (This will actually cover both of the cases of
that lemma). But $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Morley sequences
are $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}$-independent
sequences.

So $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}$ coincides
with $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Kim
independence, which is then symmetric, and
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ satisfies the
generalized freedom axiom, so $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ satisfies the
relative Kim’s lemma and $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{d}$ coincides
with Conant-independence.

######  Example 5.2.

(Random graphs without small cycles). Shelah introduces this example in Claim
2.8.5 of [27]. Let $n\geq 3$, and consider first the case where $n$ is even.
Then the theory of graphs without cycles of length not exceeding $n$ has a
model companion $T$, but it is not the model completion. The theory $T$ does
have quantifier elimination, however, in the graph language expanded by the
definable partial function symbols $F^{k}_{m}$, for $k\leq m$ not more than
$\frac{n}{2}$, sending vertices $a$ and $b$ of distance $m$ to the $k$th
vertex along the path between $a$ and $b$; note that (particularly to the even
case) any two vertices in $T$ have a unique path of length at most
$\frac{n}{2}$ between them. (We adopt the convention that paths cannot retrace
themselves.) The algebraically closed sets are then the induced subgraphs any
two vertices of which have distance within the subgraph not more than
$\frac{n}{2}$, and are determined up to elementary equivalence by their type
in the graph language. Shelah shows this theory is $\mathrm{NSOP}_{4}$.

We define a stationary relation $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ over models, as
follows. Define $a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}b$ if
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}_{M}b$ and the
algebraic closure of $Mab$ is constructed as follows, noting that this
construction is free of any choice and thus gives, by quantifier elimination,
a relation with invariance and stationarity. Let the graph given by
$\mathrm{acl}(Ma)\cup\mathrm{acl}(Mb)$ with no new edges be stage $0$. At
stage $n+1$, add formal vertices forming a unique path of length $\frac{n}{2}$
between any two vertices with no path of length at most $\frac{n}{2}$ between
them in stage $n$. Note that this cannot create small cycles, as a small cycle
can be assumed to either contain any new formal path or to not meet it at all,
but it would be too small to contain two new formal paths (and an edge from
the old graph) and if it contained only one formal path then that path would
never have been placed based on our criteria. Since $T$ is the model
companion, we have existence, and we need to show monotonicity.

To that end, we claim that if $A\subseteq B$ with $A$ algebraically closed in
$B$ (that is, each path in $B$ of length $\frac{n}{2}$ between verticies of
$A$ goes through $A$), then when we apply the construction of adding a formal
path of length $\frac{n}{2}$ between any two vertices without such a path
within $B$, we have that:

(i) the restriction of this construction to $A$ (looking only at $A$ together
with the new paths added between vertices of $A$) is the same as what we would
have obtained by looking only at $A$ originally and applying the same
construction

(ii) this restriction of the construction to $A$ remains algebraically closed
within this construction applied to $B$.

For (i), the only reason we would have added a new path between vertices of
$A$ considered alone, but not within $B$, is that there is a path between
those vertices of length $\frac{n}{2}$ within $B$ that is not within $A$. But
that does not happen, because $A$ is algebraically closed in $B$. For (ii),
any path of length at most $\frac{n}{2}$ between two nodes of the restriction
of this construction to $A$, within the construction applied to $B$, cannot go
through one of the new formal paths that are not between two vertices of $A$
as then it would be too long. So it must stay within $B$ together with the new
formal paths between vertices of $A$, but the parts of the path within $B$
will in fact be within $A$, since $A$ is algebraically closed in $B$.

It follows by induction that if $A$ is algebraically closed in $B$, then
applying all of the stages of the construction to $B$ and restricting it to
$A$ is the same as applying it to $A$ originally. But if
$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}_{M}B$ and
$A^{\prime}\subseteq A$, $B^{\prime}\subseteq B$, then
$\mathrm{acl}(MA^{\prime})\cup\mathrm{acl}(MB^{\prime})$ is algebraically
closed in $\mathrm{acl}(MA)\cup\mathrm{acl}(MB)$; for any path of length at
most $\frac{n}{2}$ between two vertices of
$\mathrm{acl}(MA^{\prime})\cup\mathrm{acl}(MB^{\prime})$, the parts within
$\mathrm{acl}(MA)$ will be in $\mathrm{acl}(MA^{\prime})$ and the parts within
$\mathrm{acl}(MB)$ will be in $\mathrm{acl}(MB^{\prime})$. Thus we have
monotonicity for $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$.

Now define the relation $A\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}_{M}^{\frac{n}{4}}B$ to mean that
$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}_{M}B$ and for
$a,b\in\mathrm{acl}(AM)\cup\mathrm{acl}(BM)$ so that there is no path of
length at most $\frac{n}{4}$ between $a$ and $b$ in the graph
$\mathrm{acl}(AM)\cup\mathrm{acl}(BM)$ with no new edges, $a$ and $b$ are of
distance greater than $\frac{n}{4}$ apart. (Note the importance of this
distance restriction in Shelah’s proof of $\mathrm{NSOP}_{4}$.) We claim that
if $M^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\frac{n}{4}}_{M}a$, then an
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Morley sequence
of copies of $a$ over $M^{\prime}$ will remain so over $M.$ To see this, note
that, if $\bigcup_{i\in I}\mathrm{acl}(M^{\prime}a_{i})$ were a disjoint union
over $M^{\prime}$ of formal copies of $\mathrm{acl}(M^{\prime}a_{i})$ over
$M^{\prime}$ with no new edges, then $\bigcup_{i\in I}\mathrm{acl}(Ma_{i})$
would be algebraically closed within $\bigcup_{i\in
I}\mathrm{acl}(M^{\prime}a_{i})$, as otherwise a path of length at most
$\frac{n}{2}$ between two of the $\mathrm{acl}(Ma_{i})$ would have to pass
through $M^{\prime}\backslash M$, and it would be too short not to pass
between one of the $\mathrm{acl}(Ma_{i})$ and $M^{\prime}\backslash M$ in no
greater than $\frac{n}{4}$ steps. So by the above observation that the
construction of repeatedly adding formal paths does not depend on the ambient
graph in which a graph is assumed algebraically closed, an
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Morley sequence
of copies of $a$ over $M^{\prime}$ will remain an
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Morley sequence
over $M$. As the same reasoning works for a set in place of $M^{\prime}$, we
see that $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{\frac{n}{4}}$
implies $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Kim
independence.

We show the reverse implication, which will tell us additionally that
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ satisfies the
relative Kim’s lemma and $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{\frac{n}{4}}$ is
Conant-independence. Suppose $a\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\frac{n}{4}}_{M}b$ is false, and
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}_{M}b$. Then
there is a path of length at most $\frac{n}{4}$ not passing through $M$
between a vertex $a$ of $\mathrm{acl}(Ma)$ and a vertex of $\mathrm{acl}(Mb)$.
Let $\varphi(x,b)\in\mathrm{tp}(a/Mb)$ imply that there is such a path. (Note
that for a path of length at most $\frac{n}{4}$ not to pass through $M$, it
need only avoid the finitely many elements of $M$ within distance
$\frac{n}{4}$ of $b$.) Suppose $\\{b_{i}\\}_{i\in\omega}$ is an invariant
Morley sequence with $\\{\varphi(x,b_{i})\\}_{i\in\omega}$ consistent,
realized by some $a^{\prime}$. Then $a^{\prime}$ will lie on a path of length
at most $\frac{n}{2}$ between vertices of
$\mathrm{acl}(Mb_{0})\cup\mathrm{acl}(Mb_{1})$ avoiding $M$, so will belong to
$\mathrm{acl}(Mb_{0}b_{1})\backslash M$. Similarly,
$a\in\mathrm{acl}(Mb_{2}b_{3})\backslash M$. But the concatenation
$\\{b_{2i}b_{2i+1}\\}_{i\in\omega}$ remains an invariant Morley sequence, so
$b_{0}b_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}_{M}^{a}b_{2}b_{3}$, a contradiction.

Note that $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{\frac{n}{4}}$
does not coincide with $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$, making this
an interesting case of Conant-independence. To see this, consider an vertex
$a$ of distance $\frac{n}{2}$ from the model $M$ and take some algebraically
closed graph (that is, a graph with no two vertices farther than $\frac{n}{2}$
apart) $B\supset M$ containing $M$ and $a$, then take two disjoint copies
$B_{1}$ and $B_{2}$ of this graph over $M$, with $a_{1}$ and $a_{2}$ the
copies of $a$ over $M$, and no further edges. Then we can add an edge of
length at most $\frac{n}{4}$ between $a_{1}$ and $a_{2}$ and not create any
small cycles. Embed this into a larger model over $M$, and the images of
$B_{1}$ and $B_{2}$ will be independent according to
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{\frac{n}{4}}$
but not $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$.

The case where $n=2m+1$ is odd is different in that, while the quantifier
elimination still holds in the language expended by the definable partial
function symbols, two vertices can be of length $m+1$ apart and none of the
partial function symbols can be defined there, in which case there are
infinitely many paths of length $m+1$ between them. So defining
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ is easier: let
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}b$ if
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}_{M}b$ and any
two vertices in $\mathrm{acl}(Ma)\cup\mathrm{acl}(Mb)$ that are not already of
distance at most $m$ apart within $\mathrm{acl}(Ma)\cup\mathrm{acl}(Mb)$ with
no new edges will have distance $m+1$. Then a similar analysis holds.

##  6\. Conant-independence in the $\mathrm{NSOP}_{n}$ hierarchy

We prove that symmetry of Conant-independence implies $\mathrm{NSOP}_{4}$. We
begin with the following fact, whose proof is essentially that of Proposition
5.2 of [21]:

######  Fact 6.1.

For formulas, Conant-forking over $M$ implies witnessing of Kim-dividing by
any Morley sequence in a type finitely satisfiable over $M$.

###### Proof.

Let $\models\varphi(x,b)\rightarrow\bigvee_{i=1}^{n}\psi_{i}(x,c_{i})$ for
$\psi_{i}(x,c_{i})$ Conant-dividing over $M$, so in particular Kim-dividing
over $M$ by any invariant Morley sequence in a finitely satisfiable type. By
left extension and monotonicity for $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{u}$ , whether or
not a formula $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-Kim-divides over
$M$ does not change when adding unused parameters, so we can assume $c_{i}=b$
for $1\leq i\leq n$. Then $\varphi(x,b)$ Kim-divides over $M$ by any invariant
Morley sequence in a finitely satisfiable type over $M$, for suppose
otherwise. Let $\\{b_{i}\\}_{i\in\mathbb{N}}$ be an invariant Morley sequence
in a finitely satisfiable type over $M$ starting with $b$; then there will be
some $a$ realizing $\\{\varphi(x,b_{i})\\}_{i\in\mathbb{N}}$. So by the
pigeonhole principle, there will be some $1\leq j\leq n$ so that $a$ realizes
$\\{\psi_{j}(x,b_{i})\\}_{i\in S}$ for $S\subseteq\mathbb{N}$ infinite. But by
monotonicity and an automorphism, we can assume $\\{b_{i}\\}_{i\in S}$ is an
invariant Morley sequence in that same finitely satisfiable type over $M$
starting with $b$, contradicting Conant-dividing of $\psi_{i}(x,b)$. ∎

The following uses similar Skolemization methods to Proposition 5.6 of
Chernikov and Ramsey in [4], which generalize in a surprising way to
indiscernible sequences ordered by a definable relation with no $4$-cycles.

######  Theorem 6.2.

Any theory where Conant-forking is symmetric is $\mathrm{NSOP}_{4}$. Thus
$n=4$ is the greatest $n$ so that there are strictly $\mathrm{NSOP}_{n}$
theories with symmetric Conant-independence.

###### Proof.

Suppose a theory $T$ has $\mathrm{SOP}_{4}$; we show that Conant-independence
cannot be symmetric. Let $R(x,y)$ be a definable binary relation with no
$4$-cycles, and let $\langle a_{i}\rangle_{i\in I}$ be an infinite sequence so
that $R(a_{i},a_{j})$ for $i<j$. Fixing a Skolemization of $T$, we can assume
that this sequence is indiscernible in that Skolemization and is of the form
$\langle c_{i}\rangle_{i\in\omega}+\langle a_{1}\rangle+\langle
b_{1}\rangle+\langle a_{2}\rangle+\langle b_{2}\rangle+\langle
a_{3}\rangle+\langle c_{i}\rangle_{i\in\omega^{*}}$. Let
$M=\mathrm{dcl}_{\mathrm{Sk}}(\langle c_{i}\rangle_{i\in\omega}+\langle
c_{i}\rangle_{i\in\omega^{*}})$, $\overline{a}=a_{1}a_{2}a_{3}$,
$\overline{b}=b_{1}b_{2}$; we show
$\overline{a}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{K^{*}}_{M}\overline{b}$ but $\overline{b}$ is Conant-dependent on
$\overline{a}$ over $M$. For the first part, clearly $\langle
c_{i}\rangle_{i\in\omega}+\langle a_{1}\rangle+\langle b_{1}\rangle+\langle
a_{2}\rangle+\langle b_{2}\rangle+\langle a_{3}\rangle+\langle
c_{i}\rangle_{i\in\omega^{*}}$ is contained in a sequence, indiscernible in
the Skolemization, of the form $\langle c_{i}\rangle_{i\in\omega}+\langle
a_{1}\rangle+\langle b^{i}_{1}\rangle_{i\in\omega}+\langle
a_{2}\rangle+\langle b_{2}^{i}\rangle_{i\in\omega^{*}}+\langle
a_{3}\rangle+\langle c_{i}\rangle_{i\in\omega^{*}}$, with $b_{j}^{0}=b_{j}$
for $j=1,2$. But $\langle b_{1}^{i}b_{2}^{i}\rangle_{i\in\omega}$ is a coheir
Morley sequence over $M$ starting with $\overline{b}$ and indiscernible over
$M\overline{a}$, so by Fact 1.1 we get
$\overline{a}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{K^{*}}_{M}\overline{b}$. For the dependent direction, we show
$R(a_{1},y_{1})\wedge R(y_{1},a_{2})\wedge R(a_{2},y_{2})\wedge
R(y_{2},a_{3})\in\mathrm{tp}(\overline{b}/M\overline{a})$ Conant-divides over
$M$. Let $\langle a^{i}_{1}a^{i}_{2}a^{i}_{3}\rangle_{i\in\omega}$ be an
$M$-invariant Morley sequence starting with $\overline{a}$ and suppose
$\\{R(a^{i}_{1},y_{1})\wedge R(y_{1},a^{i}_{2})\wedge R(a^{i}_{2},y_{2})\wedge
R(y_{2},a^{i}_{3})\\}_{i\in\omega}$ were consistent, realized by
$b^{\prime}_{1}b^{\prime}_{2}$. Then $\models
R(a^{1}_{2},b^{\prime}_{2})\wedge R(b^{\prime}_{2},a^{0}_{3})$. Now
$\models\exists xR(a^{0}_{1},x)\wedge R(x,a_{2}^{1})$, witnessed by
$b_{1}^{\prime}$. But $a_{1}^{0}\equiv_{M}a_{3}^{0}$, so by invariance,
$a_{1}^{0}\equiv_{M\overline{a}^{1}}a_{3}^{0}$, and in particular
$a_{1}^{0}\equiv_{Ma_{2}^{1}}a_{3}^{0}$. So $\models\exists
xR(a^{0}_{3},x)\wedge R(x,a_{2}^{1})$, witnessed, say, by
$b_{1}^{\prime\prime}$. But $\models R(a^{1}_{2},b^{\prime}_{2})\wedge
R(b^{\prime}_{2},a^{0}_{3})\wedge R(a_{3}^{0},b^{\prime\prime}_{1})\wedge
R(b^{\prime\prime}_{1},a_{2}^{1})$, a $4$-cycle, contradiction. ∎

Thus one of the three main classification-theoretic properties Conant proved
for free amalgamation theories in [5]–they are either $\mathrm{NSOP}_{1}$ or
$\mathrm{SOP}_{3}$, are either simple or $\mathrm{TP}_{2}$, and are
$\mathrm{NSOP}_{4}$–holds solely under the assumpton of symmetric Conant-
independence. So far, we are only able to prove the other two identities for
theories with symmetric Conant-independence and an additional assumption about
invariant types minimal in the Kim-dividing order (see section $3$); these
assumptions together generalize the free amalgamation theories and are not
known to fail in any $\mathrm{NSOP}_{4}$ theory, but it would be desirable if
we had a criterion analogous to independence in the simple or
$\mathrm{NSOP}_{1}$ case that gave us all of the classification-theoretic
properties of free amalgamation theories. Can we get those other two
properties with just symmetry for Conant-independence alone?

######  Problem 6.3.

Must a theory with symmetric Conant-independence be either simple or
$\mathrm{TP}_{2}$? Must it be either $\mathrm{NSOP}_{1}$ or
$\mathrm{SOP}_{3}$?

We are also interested in extending the theory of Kim-independence beyond
$\mathrm{NSOP}_{1}$. Given that the class of strictly $\mathrm{NSOP}_{4}$
theories is the most complicated classification-theoretic class where Conant-
independence is symmetric, we may ask whether symmetry for Conant-independence
characterizes $\mathrm{NSOP}_{4}$ the same way symmetry for Kim-independence
characterizes $\mathrm{NSOP}_{1}$.

######  Problem 6.4.

In an $\mathrm{NSOP}_{4}$ theory, is Conant-independence always symmetric?

A positive answer to both the last problem and one of the two questions from
the previous problem will solve some of the open regions of the
classification-theoretic hierarchy, further underscoring the connections
between classification theory and the theory of model-theoretic independence.

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