# An $\mathrm{NSOP}_{1}$ theory without the existence axiom

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

Answering a question of Dobrowolski, Kim and Ramsey, we find an
$\mathrm{NSOP}_{1}$ theory that does not satisfy the existence axiom.

This work was supported by the NSF under Grant No. DMS-2303034.

##  1\. Introduction

One of the core informal questions of model theory is to determine what role
stability theory, famously introduced by Morley ([19]) and Shelah ([23]) to
classify the number of non-isomorphic models (of a given size) of a first-
order theory, can play in describing theories that are themselves unstable–and
that may not even be simple. This project was initiated in large part in
celebrated work of Kim ([15]) and Kim and Pillay ([17]), who showed that the
(forking-)independence relation has many of the same properties in simple
theories that it does in stable theories, and in fact that these properties
characterize simplicity. In a key step towards the non-simple case, Kaplan and
Ramsey ([12]) then defined Kim-independence over models, which generalizes the
definition of forking-independence:

######  Definition 1.1.

Let $M\models T$. A formula $\varphi(x,b)$ _Kim-divides_ over $M$ if there is
an $M$-invariant Morley sequence $\\{b_{i}\\}_{i\in\omega}$ starting with $b$
such that $\\{\varphi(x,b_{i})\\}_{i\in\omega}$ is inconsistent. A formula
$\varphi(x,b)$ _Kim-forks_ over $M$ if it implies a (finite) disjunction of
formulas Kim-dividing over $M$. We write $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}^{K}_{M}b$, and
say that $a$ is _Kim-independent_ from $b$ over $M$, if $\mathrm{tp}(a/Mb)$
does not include any formulas Kim-forking over $M$.

In, for example, [6], [12], [13], [14], it is shown that Kim-independence has
many of the same properties in theories without the first strict order
property–$\mathrm{NSOP}_{1}$ theories–that forking-independence has in simple
theories. It is also shown that $\mathrm{NSOP}_{1}$ theories have a
characterization in terms of properties of Kim-independence, similarly to how
simplicity has a characterization in terms of properties of forking-
independence. However, in contrast to the case of simple theories, this work
only describes the properties of a relation $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}^{K}b$
defined when $M$ is a model, leaving open the question of whether independence
phenomena in $\mathrm{NSOP}_{1}$ theories extend from independence over models
to independence over sets.

The following axiom of first-order theories, though introduced earlier under
different names (see, e.g. [5]), was defined by Dobrowolski, Kim and Ramsey in
[9]:

######  Definition 1.2.

A theory $T$ satisfies the existence axiom if no type $p\in S(A)$ forks over
$A$.

This is equivalent to every type $p\in S(A)$ having a global extension that
does not fork over $A$. All simple theories satisfy the existence axiom
([15]), while the circular ordering is an example of a dependent theory not
satisfying the existence axiom ([16], Example 2.11). In [9] it is shown that,
in $\mathrm{NSOP}_{1}$ theories satisfying the existence axiom, it is possible
to extend the definition of Kim-independence $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{C}^{K}b$ from
the case where $C$ is a model to the case where $C$ is an arbitrary set, so
that the relation $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}^{K}b$ will have similar properties to Kim-
independence over models in general $\mathrm{NSOP}_{1}$ theories.
Specifically, because, under the existence axiom, (nonforking-)Morley
sequences are defined over arbitrary sets, they can be used in place of
invariant Morley sequences111It is easy to see that invariant Morley sequences
are not defined over arbitrary sets in $\mathrm{NSOP}_{1}$ theories: for
example, any set $A$ such that $\mathrm{acl}(A)\neq\mathrm{dcl}(A)$. to define
Kim-dividing of a formula over a set:

######  Definition 1.3.

([9]) Let $T$ be a $\mathrm{NSOP}_{1}$ theory, and let $C\subset\mathbb{M}$ be
an arbitrary set. A formula $\varphi(x,b)$ _Kim-divides_ over $C$ if there is
a (nonforking-)Morley sequence $\\{b_{i}\\}_{i\in\omega}$ over $C$ starting
with $b$ such that $\\{\varphi(x,b_{i})\\}_{i\in\omega}$ is inconsistent. A
formula $\varphi(x,b)$ _Kim-forks_ over $C$ if it implies a (finite)
disjunction of formulas Kim-dividing over $C$. We write
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{A}b$, and say that $a$ is _Kim-
independent_ from $b$ over $C$, if $\mathrm{tp}(a/Ab)$ does not include any
formulas Kim-forking over $C$.

Dobrowolski, Kim and Ramsey ([9]) show that, in an $\mathrm{NSOP}_{1}$ theory
satisfying the existence axiom, the Kim-independence relation
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}$ as defined above over arbitrary sets
satisfies symmetry and the independence theorem (for Lascar strong types), and
that Kim-dividing over arbitrary sets satisfies Kim’s lemma and coincides with
Kim-forking. Chernikov, Kim and Ramsey, in [2], show even more properties of
Kim-independence over arbitrary sets in $\mathrm{NSOP}_{1}$ theories
satisfying the existence axiom, including transitivity and witnessing. (All of
these properties correspond to the properties of Kim-independence over models
proven in general $\mathrm{NSOP}_{1}$ theories in [12], [13].) Motivated by
these results, [9], and later [3], ask:

######  Question 1.4.

Does every $\mathrm{NSOP}_{1}$ theory satisfy the existence axiom?

We show that the answer to this question is no:

######  Theorem 1.5.

There is an $\mathrm{NSOP}_{1}$ theory not satisfying the existence axiom.

This contrasts with the work of Kim, Kim and Lee in [3], where, for the
definition of Kim-forking over sets given by Dobrowolski, Kim and Ramsey in
[9] (Defintion 1.3 above), it is shown that no type $p\in S(A)$ Kim-forks over
$A$ (i.e. contains a formula Kim-forking over $A$), where $A$ is an arbitrary
set in an $\mathrm{NSOP}_{1}$ theory. In fact, our example is the first known
example of a theory without the strict order property, or $\mathrm{NSOP}$
theory, not satisfying the existence axiom. Prior to our results, the theory
of an algebraically closed field with a generic multiplicative endomorphism,
constructed by d’Elbée in [8], was suggested there as a candidate for an
$\mathrm{NSOP}_{1}$ theory without the existence axiom; whether the theory
constructed by d’Elbée satisfies the existence axiom was left unresolved in
that article. However, as discussed in a personal communication with d’Elbée
([7]), it is expected that that theory, which was shown in [8] to be
$\mathrm{NSOP}_{1}$, actually does satisfy the existence axiom.

Our construction is based on the theory of $\omega$-stable free pseudoplanes,
a classical example of a non-one-based $\omega$-stable theory with trivial
forking discussed in, say, [20]. The theory of $\omega$-stable free
pseudoplanes is the theory of undirected graphs of infinite minimum degree
without cycles. Variations of this construction appear in e.g. [10], [1],
[24], [11]. Note also that some arguments from the below, including the
strategy, in axiom schema $T_{4}$, of requiring that $n$ connected sets of
pairwise distance greater than $2^{n}$ can be colored independently, and the
associated Claims 2.1 and 2.2, their proofs, and their application in proving
claim (***), are formally similar to those of Section 3 of Chernikov,
Hrushovski, Kruckman, Krupiński, Moconja, Pillay and Ramsey ([4]).

##  2\. The construction

Let $\mathcal{L}$ be the language with sorts $P$ and $O$, symbols $R_{1}$ and
$R_{2}$ for binary relations on $O$, and symbols $\rho_{1}$ and $\rho_{2}$ for
binary relations between $P$ and $O$. Call an $\mathcal{L}$-structure $A$
copacetic if:

(C1) For $i=1,2$, $R_{i}(A)$ is a symmetric, irreflexive relation on $O(A)$,
and the two are mutually exclusive: for $a_{1},a_{2}\in O(A)$, $A\not\models
R_{1}(a_{1},a_{2})\wedge R_{2}(a_{1},a_{2})$.

(C2) The relation $R_{1}(A)\cup R_{2}(A)$ has no loops on $O(A)$ (i.e. there
are no distinct $a_{0}\ldots a_{n-1}\in O(A)$, $n>2$, and $i_{1}\ldots
i_{n}\in\\{1,2\\}$ so that, for $0\leq j\leq n-1$, $A\models
R_{i_{j}}(a_{i},a_{i+1\mathrm{\>mod\>}n})$).

(C3) For all $b\in P(A)$, $a\in O(A)$, exactly one of $A\models\rho_{1}(b,a)$
and $A\models\rho_{2}(b,a)$ hold.

(C4):

(a) For each $b\in P(A)$, there are no distinct $a_{1},a_{2}\in O(A)$, so that
there there is some $a_{*}\in O(A)$ so that $A\models R_{1}(a_{1},a_{*})\wedge
R_{1}(a_{2},a_{*})$, and $A\models\rho_{1}(b,a_{1})\wedge\rho_{1}(b,a_{2})$.

(b) For each $b\in P(A)$, there are no distinct $a_{1},a_{2},a_{3}\in O(A)$,
so that there there is some $a_{*}\in O(A)$ so that $A\models
R_{2}(a_{1},a_{*})\wedge R_{2}(a_{2},a_{*})\wedge R_{2}(a_{3},a_{*})$, and
$A\models\rho_{2}(b,a_{1})\wedge\rho_{2}(b,a_{2})\wedge\rho_{2}(b,a_{3})$.

Let $A$ be copacetic, and let $b\in P(A)$, $a\in O(A)$, $i\in{1,2}$. Then
there are at most $i$ many $a^{\prime}\in O(A)$ with $A\models
R_{i}(a,a^{\prime})$ and $A\models\rho_{i}(b,a^{\prime})$. For $N$ the number
of such $a^{\prime}$ that are defined, let $b^{A,j}_{\to^{i}}(a)$, $1\leq
j\leq N$, denote the $N$ many such $a^{\prime}$ (so if $i=2$ and there are two
such $a^{\prime}$, make an arbitrary choice of which is $b^{A,1}_{\to^{i}}(a)$
and which is $b^{A,2}_{\to^{i}}(a)$, while if $i=1$, then
$b^{A,1}_{\to^{i}}(a)$ is the sole such $a^{\prime}$, if it exists.)

If $B$ is copacetic and $A\subset B$ is a substructure of $B$ (and is
therefore also copacetic), call $A$ closed in $B$ (denoted $A\leq B$) if

(i) For $b\in P(A)$, $a\in O(A)$, $i\in\\{1,2\\}$, $1\leq j\leq i$, if
$b^{B,j}_{\to^{i}}(a)$ exists, then $b^{B,j}_{\to^{i}}(a)\in A$.

(ii) Any $R_{1}(B)\cup R_{2}(B)$-path between nodes of $O(A)$ lies in $O(A)$:
for all $a_{1},a_{n}\in O(A)$, $a_{2},\ldots a_{n-1}\in O(B)$ which are
distinct and distinct from $a_{1}$, $a_{n}$, and
$i_{1},\ldots,i_{n-1}\in\\{1,2\\}$, if $A\models R_{i_{j}}(a_{i},a_{i+1})$ for
$1\leq j\leq n-1$, then for all $2\leq i\leq n-1$, $a_{i}\in O(A)$.

Call a copacetic $\mathcal{L}$-structure $A$ connected if $R_{1}(A)\cup
R_{2}(A)$ forms a connected graph on $O(A)$: for all $a,a^{\prime}\in O(A)$,
there are $a_{2},\ldots a_{n-1}\in O(A)$, $i_{1},\ldots,i_{n-1}\in\\{1,2\\}$
for some $n$, so that for $a_{1}=a$, $a_{n}=a^{\prime}$, $A\models
R_{i_{j}}(a_{i},a_{i+1})$ for all $1\leq j\leq n-1$. So if $B$ is copacetic
and $A\subseteq B$, connectedness of $A$ supplants requirement (ii) of $A$
being closed in $B$. Call a subset of $A$ a connected component of $A$ if it
is a maximal connected subset of $O(A)$.

Let $O$ be an undirected graph without cycles and with a $2$-coloring of its
edges, with $R_{1}$, $R_{2}$ denoting edges of either color. Let
$\rho_{1},\rho_{2}\subset O$, $O=\rho_{1}\cup\rho_{2}$,
$\rho_{1}\cap\rho_{2}=\emptyset$ be a coloring of the vertices of $O$ so that,
for $i=\\{1,2\\}$, no $i+1$ distinct vertices of $O$, lying on the boundary of
the same $R_{i}$-ball of radius $1$ (i.e. they have a common
$R_{i}$-neighbor), are both colored by $\rho_{i}$. Then we call
$\rho_{1},\rho_{2}$ a (C4)-coloring of $O$. For $A$ copacetic,
$O^{\prime}\subseteq O(A)$, and $\rho_{1},\rho_{2}\subseteq O^{\prime}$ a
(C4)-coloring of $O^{\prime}$, say that $b\in P(A)$ induces the (C4)-coloring
$\rho_{1},\rho_{2}$ on $O^{\prime}$ if for $i\in\\{1,2\\}$, $\rho_{i}=\\{a\in
O^{\prime}:A\models\rho_{i}(b,a)\\}$.

The following assumptions on an $\mathcal{L}$-structure $A$ are expressible by
a set of first-order sentences:

($T_{1}$) (Copaceticity) $A$ is copacetic.

($T_{2}$) (Completeness) For $b\in P(A)$, $a\in O(A)$, $i\in\\{1,2\\}$, if
$A\models\neg\rho_{i}(b,a)$, then $b^{A,j}_{\to^{i}}(a)$ exists for $1\leq
j\leq i$.

($T_{3}$) (Tree extension) For $C\subseteq A$, and $B\geq C$ finite and
copacetic with $P(B)=P(C)$, there is an embedding $\iota:B\hookrightarrow A$
with $\iota|_{C}=\mathrm{id}_{C}$.

($T_{4}$) (Parameter introduction) For any $n<\omega$, finite connected sets
$O_{1},\ldots,O_{n}\subseteq O(A)$ so that there does not exist an $R_{1}\vee
R_{2}$-path of length at most $2^{n}$ between a vertex of $O_{i}$ and a vertex
of $O_{j}$ for any distinct $i,j\leq n$ (so in particular, $O_{i}$ and $O_{j}$
are disjoint), and (C4)-colorings $\rho^{i}_{1},\rho^{i}_{2}\subseteq O_{i}$
of $O_{i}$ for $i\leq n$, there are infinitely many $b\in P(A)$ so that, for
each $i\leq n$, $b$ induces the (C4)-coloring
$\rho^{i}_{1},\rho^{i}_{2}\subseteq O_{i}$ on $O_{i}$.

Let $T^{\not\exists}=T_{1}\cup T_{2}\cup T_{3}\cup T_{4}$. We claim that
$T^{\not\exists}$ is consistent. This will follow by induction from the
following three claims:

(*) If $A$ is copacetic, $b\in P(A)$, $a\in O(A)$, $i\in\\{1,2\\}$, and $1\leq
j\leq i$, there is a copacetic $\mathcal{L}$-structure $A^{\prime}\supseteq A$
(i.e. containing $A$ as a substructure) so that
$b^{A^{\prime},j}_{\to^{i}}(a)$ exists.

(**) If $A$ is copacetic, $C\subseteq A$, and $B\geq C$ is finite and
copacetic with $P(B)=P(C)$, there is a copacetic structure
$A^{\prime}\supseteq A$ and an embedding $\iota:B\hookrightarrow A^{\prime}$
with $\iota|_{C}=\mathrm{id}_{C}$.

(***) If $A$ is copacetic, for any $n<\omega$, finite connected sets
$O_{1},\ldots,O_{n}\subseteq O(A)$ so that there does not exist an $R_{1}\vee
R_{2}$-path of length at most $2^{n}$ between a vertex of $O_{i}$ and a vertex
of $O_{j}$ for any distinct $i,j\leq n$, and (C4)-colorings
$\rho^{i}_{1},\rho^{i}_{2}\subseteq O_{i}$ of $O_{i}$ for $i\leq n$, there is
some copacetic $A^{\prime}\supset A$ and $p\in P(A^{\prime}\backslash A)$ so
that, for each $i\leq n$, $p$ induces the (C4)-coloring
$\rho^{i}_{1},\rho^{i}_{2}$ on $O_{i}$.

We first show (*). Suppose $b^{A,j}_{\to^{i}}(a)$ does not already exist. Let
$A^{\prime}=A\cup\\{*\\}$ for $*$ a new point of sort $O$, and let
$R_{i}(A^{\prime})=R(A)\cup\\{(a,*),(*,a)\\}$,
$R_{3-i}(A^{\prime})=R_{3-i}(A)$,
$\rho_{3-i}(A^{\prime})=\rho_{3-i}(A)\cup(P(A)\backslash\\{b\\})\times\\{*\\}$,
$\rho_{i}(A^{\prime})=\rho_{i}(A)\cup\\{(b,*)\\}$. Then $A^{\prime}$ is
copacetic; first, (C1)-(C3) are clearly satisfied. Second, no point of
$P(A)\backslash\\{b\\}$ can witness a failure of (C4), because $A$ is
copacetic and $*$ is not an $R_{3-i}$-neighbor of any point of
$O(A^{\prime})$. Finally, neither can $b$ witness a failure of (C4), because
$*$ is not an $R_{i-3}$ neighbor of any point of $O(A^{\prime})$ and $a$ is
the unique $R_{i}$-neighbor of $*$ in $O(A^{\prime})$, that failure must by
copaceticity of $A$ be witnessed on the boundary of the $R_{i}$-ball of radius
$1$ centered at $a$. But because $b^{A,j}_{\to^{i}}(a)$ does not already
exist, there are fewer than $i$ many $R_{i}$-neighbors $a^{\prime}$ of $a$ in
$O(A)$ with $A\models\rho_{i}(b,a^{\prime})$, so there are at most $i$ many
$R_{i}$-neighbors $a^{\prime}$ of $A$ in $O(A^{\prime})$ with
$A\models\rho_{i}(b,a^{\prime})$; thus the failure of (C4) is not in fact
witnessed on this ball’s boundary. By construction, we can then choose
$b^{A^{\prime},j}_{\to^{i}}(a)=*$.

We next show (**). For this we need the following claim:

######  Claim 2.1.

Let $O^{\prime}$ be undirected graph without cycles and with a $2$-coloring of
its edges, with $R_{1}$, $R_{2}$ denoting edges of either color. Let $O\subset
O^{\prime}$ be a subgraph with the induced coloring, and with $O\leq
O^{\prime}$ (in the sense of (ii), so any path between two vertices of
$O^{\prime}$ consisting of edges of any colors is contained in $O$). Then any
(C4)-coloring $\rho_{1},\rho_{2}$ of $O$ extends to a (C4)-coloring
$\rho_{1}^{\prime},\rho^{\prime}_{2}$ of $O^{\prime}$, which has the following
additional property: if $v\in O^{\prime}\backslash O$ has an $R_{i}$-neighbor
in $O$, then $v\in\rho^{\prime}_{3-i}$.

###### Proof.

By the assumption on paths in $O^{\prime}$ between two vertices of $O$, we may
decompose $O^{\prime}\backslash O$ into a disjoint union $\sqcup O^{i}$ of
connected subgraphs, so that each $O^{i}$ has at most one vertex $v_{i}$ with
any neighbors in $O$; $v_{i}$ will in fact have only one neighbor $w_{i}$ in
$O$. For every $O^{i}$ all of whose vertices have no neighbors in $O$, let
$v_{i}$ be an arbitrary vertex of $O^{i}$. Then inductively, we can order each
$O^{i}$ as a tree (i.e., a partial order with linearly ordered downsets) so
that $v_{i}$ is the root, any node’s immediate successors are all neighbors of
that node and, among any two neighbors, one must be an immediate successor of
the other, and each maximal linearly ordered set is well-ordered of order type
at most $\omega$. Extend $\rho_{1},\rho_{2}$ on each $O_{i}$ as follows,
starting from $v_{i}$ and proceeding by induction. If $w_{i}$ is an
$R_{i}$-neighbor of $v_{i}$, color $v_{i}$ by $\rho^{\prime}_{3-i}$;
otherwise, color $v_{i}$ arbitrarily. Then for each vertex $v$ of $O^{i}$,
each immediate successor of $v$ will be an $R_{i}$-neighbor of $v$ for some
$i\in\\{1,2\\}$; color it by $\rho_{3-i}$. No two distinct vertices $v,w$ with
an $R_{i}$-distance of exactly $2$ in $O^{\prime}$ can then be colored in
$\rho^{i}$: if the $R_{i}$-path between $v$ and $w$ goes through $O$, then
whichever of $v$ and $w$ is not in $O$ will be colored by
$\rho^{\prime}_{3-i}$, and if the $R_{i}$-path between $v$ and $w$ stays in
$O^{i}$, whichever of $v$ and $w$ is not the least in the path will be colored
by $\rho^{\prime}_{3-i}$. This shows that
$\rho^{\prime}_{1},\rho^{\prime}_{2}$ is a (C4)-coloring, and the additional
property is immediate from the construction.

∎

Now, take the set $A^{\prime}$ to be the disjoint union of $A$ and $B$ over
$C$; for notational simplicity we identify $A$, $B$ and $C$ respectively with
their images in $A^{\prime}$. Let the $R_{1},R_{2}$-structure on
$O(A^{\prime})$ be given as follows: the identification on $O(A)$ and $O(B)$
preserves the $R_{i}$-structure, and $O(A)$ and $O(B)$ are freely amalgamated
over $O(C)$ (i.e. there are no $R_{i}$-edges between $O(A\backslash C)$ and
$O(B\backslash C)$). This guarantees (C1), and also guarantees (C2) by
condition (ii) of $C\leq B$. Now let the $\rho_{1},\rho_{2}$-structure on
$A^{\prime}$ be given as follows: the identifications on $A$ and $B$ preserve
the $\rho_{i}$-structure, which, by the assumption that $P(B)=P(C)$, leaves us
only by way of satisfying (C3) to define the $\rho_{1},\rho_{2}$-structure on
$P(A\backslash C)\times O(B\backslash C)$, and this will be the following. Let
$p\in P(A\backslash C)$; the requirement that the $\rho_{i}$-structure on $A$
is preserved tells us the (C4)-coloring that $p$ induces on $O(C)$, and we
extend this to a (C4)-coloring on $O(B)$ as in Claim 2.1; here we use
condition (ii) of $C\leq B$. We have defined the full $\mathcal{L}$-structure
on $A^{\prime}$, so it remains to show (C4); in other words, we show that $p$
induces a (C4)-coloring on $A^{\prime}$ in the case where $p\in P(C)$ and in
the case where $p\in P(A\backslash C)$. In either case $p$ induces a
(C4)-coloring on $A$ and on $B$, so the only way (C4) can fail is on the
boundary of an $R_{i}$-ball of radius $1$ centered at $c\in C$. But (C4)
cannot fail on the boundary of this ball, because it does not fail there in
$A$, and by condition (i) of $C\leq B$ in the first case, or by the additional
clause of Claim 2.1 in the second case, any $R_{i}$-neighbor of $c$ in
$B\backslash C$ is colored by $\rho_{3-i}$ in the (C4)-coloring induced by $p$
on $B$, so (C4) still cannot fail on the boundary of this ball in
$A^{\prime}$. So $A^{\prime}$ is a copacetic $\mathcal{L}$-structure
containing $A$, and clearly, the embedding $\iota$ as in (**) exists, so this
shows (**).

Finally, we show (***). For this, we need an additional combinatorial claim:

######  Claim 2.2.

Let $O$ be be undirected graph without cycles and with a $2$-coloring of its
edges, with $R_{1}$, $R_{2}$ denoting edges of either color, and let
$O_{1},\ldots,O_{n}\subseteq O$ be connected subsets so that there does not
exist an $R_{1}\vee R_{2}$-path of length at most $2^{n}$ between a vertex of
$O_{i}$ and a vertex of $O_{j}$ for any distinct $i,j\leq n$. Let
$\rho^{i}_{1},\rho^{i}_{2}\subseteq O_{i}$ be a (C4)-coloring of $O_{i}$ for
$i\leq n$. Then there is some $O^{\prime}\leq O$ containing
$O_{1}\cup\ldots\cup O_{n}$ and (C4)-coloring $\rho_{1},\rho_{2}$ of $O$ so
that, for $i\leq n$, $\rho_{1},\rho_{2}$ restricts to
$\rho^{i}_{1},\rho^{i}_{2}$ on $O_{i}$.

###### Proof.

Considering each connected component of $O$ individually, we may assume $O$ to
be connected. We prove this claim by induction on $n$. Because we will later
consider a variant of this construction where the case $n=2$ is the main
difference, we isolate this case, which is necessary for the induction, as a
subclaim:

######  Subclaim 2.3.

Claim 2.2 is true where $O$ is connected and $n=2$.

###### Proof.

Let $I$ be the shortest path between $O_{1}$ to $O_{2}$, which will consist,
ordered in the direction from $O_{1}$ to $O_{2}$, of $o_{0},\ldots o_{n}$, for
$o_{0}\in O_{1}$, $o_{n}\in O_{2}$, $o_{1}\ldots o_{n-1}\in
O\backslash(O_{1}\cup O_{2})$, and $n\geq 5$. Because $O_{1}$ and $O_{2}$ are
connected, it suffices to color $O_{1}\cup O_{2}\cup I$ so as to extend the
colorings $\rho^{i}_{1},\rho^{i}_{2}$ on $O_{i}$ for $i=1,2$, and to preserve
the condition of being a (C4)-coloring. So color $O_{1}$ by
$\rho^{1}_{1},\rho^{1}_{2}$, $O_{2}$ by $\rho^{2}_{1},\rho^{2}_{2}$, and color
$o_{1}\ldots o_{n-1}$ as follows: first, color $o_{1}$ by $\rho_{3-i}$, for
$i$ so that $o_{0}$ is an $R_{i}$-neighbor of $o_{1}$, and color $o_{n-1}$ by
$\rho_{3-i}$, so that $o_{n-1}$ is an $R_{i}$-neighbor of $o_{n}$. Since we
colored $O_{1}$ by $\rho^{1}_{1},\rho^{1}_{2}$ and $O_{2}$ by
$\rho^{2}_{1},\rho^{2}_{2}$, the only vertices $o$ of the $O_{i}$ so that the
condition of being a (C4)-coloring can fail in $O_{1}\cup O_{2}\cup I$ on the
boundary of the $R_{i}$-ball of radius $1$ centered at $o$ are $o_{0}$ and
$o_{n}$, and we have just prevented this. It remains to color $o_{2},\ldots
o_{n-2}$; we color them all by $\rho_{2}$. The only remaining vertices $o$ of
$O_{1}\cup O_{2}\cup I$ where the condition of being a (C4)-coloring can fail
on the $1$-ball centered at $O$ are now $o_{1}\ldots o_{n-1}$, but the
boundaries of the $1$-balls centered at those vertices in $O_{1}\cup O_{2}\cup
I$ have just two points, at least one of which must be colored by $\rho_{2}$,
because $n\geq 5$ and the interval $\\{o_{2},\ldots o_{n-2}\\}$ whose points
we colored by $\rho_{2}$ contains at least one of the two neighbors of each
$o_{1},\ldots,o_{n-1}$. So the condition of being a (C4)-coloring cannot fail
there. ∎

We now consider the general case of the induction. Without loss of generality,
we may assume that $O_{n-1}$ and $O_{n}$ have minimum distance $d>2^{n}$ among
any $O_{i},O_{j}$ with $i,j$ distinct. By Lemma 2.3, we may find a
(C4)-coloring $(\rho^{\prime})^{n-1}_{1},(\rho^{\prime})^{n-1}_{2}$ of
$O^{\prime}_{n-1}=:O_{n-1}\cup O_{n}\cup I$ extending
$\rho^{n-1}_{1},\rho^{n-1}_{2}$ and $\rho^{n}_{1},\rho^{n}_{2}$, where $I$ is
the shortest path, which will be of length $d$, between $O_{n-1}$ and $O_{n}$.
Clearly $O^{\prime}_{n-1}$ is connected, so it suffices to show that no two of
$O_{1},\ldots O_{n-2},O^{\prime}_{n-1}$ have distance less than
$\frac{d}{2}>2^{n-1}$, because then we can apply the inductive step. Let
$i\leq n-2$; it suffices to show that $O_{i}$ does not have distance less than
$\frac{d}{2}$ from $O^{\prime}_{n-1}$. Suppose otherwise; then it has distance
less than $\frac{d}{2}$ from some point $p$ of $O^{\prime}_{n-1}$, but by
definition of $O^{\prime}_{n-1}$, $p$ must have distance at most $\frac{d}{2}$
from either $O_{n-1}$ or $O_{n}$. So $O_{i}$ must have distance less than
$\frac{d}{2}+\frac{d}{2}=d$ from either $O_{n-1}$ or $O_{n}$, contradicting
minimality of $d$. ∎

By Claims 2.2 and 2.1, there is a (C4)-coloring $\rho_{1},\rho_{2}$ on $O(A)$
extending the (C4)-coloring $\rho^{i}_{1},\rho^{i}_{2}$ on $O_{i}$ for each
$i\leq n$. So we can extend $A$ to $A^{\prime}=A\cup p$, where $p\in
P(A^{\prime}\backslash A)$, and define the relations $\rho_{1}$ and $\rho_{2}$
on $O(A)\times\\{p\\}$ so that $p$ induces the (C4)-coloring
$\rho_{1},\rho_{2}$ on $O(A)$. This proves (***), and the consistency of
$T^{\not\exists}$.

We call a copacetic $\mathcal{L}$-structure complete if it satisfies axiom
$T_{2}$; completeness of $A$ will always supplant condition (i) of $A\leq B$.

We now prove that $T^{\not\exists}$ has the following embedding property:

######  Lemma 2.4.

Let $\mathbb{M}$ be a sufficiently saturated model of $T^{\not\exists}$. Let
$C\leq\mathbb{M}$, and let $B\geq C$ be a small copacetic
$\mathcal{L}$-structure. Assume additionally that $B$ is complete. Then there
is an embedding $\iota:B\hookrightarrow\mathbb{M}$ that is the identity on $C$
and satisfies $\iota(B)\leq\mathbb{M}$.

###### Proof.

We prove the lemma in the following two cases:

(1) $P(B)=P(C)$

(2) $O(B)=O(C)$ and $|P(B\backslash C)|=1$.

These cases suffice, because the closed extension $C\leq B$ can be decomposed
into a closed extension satisfying (1) followed by an ascending chain of
(obviously closed) extensions satisfying (2), and the property
$\iota(B)\leq\mathbb{M}$ is clearly preserved under taking unions.

To prove case (1), by completeness of $B$ and saturatedness of $\mathbb{M}$,
it suffices to find, for arbitrarily large $n<\omega$, an embedding
$\iota_{n}:B\hookrightarrow\mathbb{M}$ that is the identity on $C$ and so that
points of $\iota_{n}(B)$ that are not connected by a path in $\iota_{n}(B)$
have distance at least $n$ in $\mathbb{M}$. We claim that, for every copacetic
$B^{\prime}\geq C$, and any points $b,b^{\prime}\in O(B)$ so that $b$ and
$b^{\prime}$ belong to distinct connected components of $B$ and $b^{\prime}$
belongs to a connected component of $B$ not containing any point of $C$, there
is some copacetic $B^{\prime\prime}\supseteq B^{\prime}$ so that $C\leq
B^{\prime\prime}$, that consists of $B^{\prime}$ together with a path of from
$b$ to $b^{\prime}$ of length greater than $n$. One way to do this is to add
an $R_{1}$-path of length greater than $n$ (and greater than $2$) between $b$
and $b^{\prime}$, so that all induced (C4)-colorings color the new nodes of
this path by $\rho_{2}$. By choice of induced colorings, the resulting
$\mathcal{L}$-structure $B^{\prime\prime}$ will satisfy (C4) and condition (i)
of $C\leq B$, and $B^{\prime\prime}$ will also satisfy condition (ii) of
$C\leq B$ by the assumption of what connected components of $B^{\prime\prime}$
have $b$ and $b^{\prime}$ as members. By repeatedly applying this claim to
connect each connected component of $B$ not meeting $C$ to some fixed
connected component of $B$ meeting $C$ (or an arbitrary fixed connected
component of $B$, if $C$ is empty), we then obtain a copacetic
$\mathcal{L}$-structure $B^{\prime}\supseteq B$ with $P(B^{\prime})=P(C)$ and
$B^{\prime}\geq C$ such that, for any $b,b^{\prime}\in B$ that are not
connected in $B$, either $b$ and $b^{\prime}$ have finite distance greater
than $n$ in $B^{\prime}$, or they belong to different connected components of
$B^{\prime}$ meeting $C$. Now suppose
$\iota^{\prime}:B^{\prime}\hookrightarrow\mathbb{M}$ is any embedding
restricting to the identity on $C$; then for
$\iota^{\prime}(b),\iota^{\prime}(b^{\prime})\in\iota^{\prime}(B)$, that are
not connected in $\iota^{\prime}(B)$, either $b$ and $b^{\prime}$ have finite
distance greater than $n$ in $B^{\prime}$, so $\iota(b)$ and
$\iota(b^{\prime})$ have distance greater than $n$ in $\mathbb{M}$ as desired,
or $b$ and $b^{\prime}$ belong to different connected components of
$B^{\prime}$ meeting $C$, so $\iota(b)$ and $\iota(b^{\prime})$ belong to
different connected components of $\iota(B^{\prime})$ meeting $C$, so are not
connected in $\mathbb{M}$ because $C\leq\mathbb{M}$; then
$\iota_{n}=\iota^{\prime}|_{B}$ is as desired. So if we can show that, for any
small, copacetic $\mathcal{L}$-structure $B^{\prime}\geq C$ with
$P(B^{\prime})=P(C)$, there is an embedding $\iota:B^{\prime}\to\mathbb{M}$
that is the identity on $C$ (with no additional requirements on $\iota(B)$),
we will have proven case (1). Note that if $B_{0}\subseteq B^{\prime}$,
$B_{0}\cap C\leq B_{0}$, so we may assume that $B^{\prime}$ is finite by
saturatedness of $\mathbb{M}$. Then we can just apply tree extension.

To prove case (2), let $\rho_{1},\rho_{2}$ be the (C4)-coloring induced by
$p$, where $\\{p\\}=P(B\backslash C)$, on $C$. Since $B$ is complete and
$C\leq\mathbb{M}$, any embedding $\iota:B\to\mathbb{M}$ with
$\iota|_{C}=\mathrm{id}$ will satisfy $\iota(B)\leq\mathbb{M}$. By
saturatedness of $\mathbb{M}$ it suffices to find, for every finite
$P_{0}\subset P(C)$ and finite $C_{0}\subset O(C)$, some
$p\in\mathbb{M}\backslash P_{0}$ inducing the (C4)-coloring
$\rho_{1}|_{C_{0}},\rho_{2}|_{C_{0}}$ on $C_{0}$. We may enlarge $C_{0}$ to a
finite subset of $O(C)$ consisting of a finite union of $n$ connected sets
that are not connected in $C$, so by condition (ii) of $C\leq\mathbb{M}$, are
not connected in $\mathbb{M}$; in particular, any two of them have distance
greater than $2^{n}$ in $\mathbb{M}$. So by parameter introduction, we may
find infinitely many $p\in P(\mathbb{M})$ inducing the (C4)-coloring
$\rho_{1}|_{C_{0}},\rho_{2}|_{C_{0}}$ on $C_{0}$, so in particular, one not
belonging to $P_{0}$.

∎

We now fix a sufficiently saturated $\mathbb{M}\models T^{\not\exists}$, which
we take as the ambient model; for $p\in P(\mathbb{M})$, fix the notation,
$p^{j}_{\to^{i}}=:p^{\mathbb{M},j}_{\to^{i}}$ The following quantifier
elimination is a corollary of the above:

######  Corollary 2.5.

Let $A,B\leq\mathbb{M}$. Then if
$\mathrm{qftp}_{\mathcal{L}}(A)=\mathrm{qftp}_{\mathcal{L}}(B)$,
$\mathrm{tp}(A)=\mathrm{tp}(B)$.

###### Proof.

For any $A\leq\mathbb{M}$ and $a\in\mathbb{M}$, there is some (small)
$B\leq\mathbb{M}$ so that $Aa\subset B$, and in particular $A\leq B$. Because
$B\leq\mathbb{M}$ and $\mathbb{M}$ satisfies completeness, $B$ is complete. So
the corollary follows by Lemma 2.4 by a back-and-forth argument. ∎

Next, we show that $T^{\not\exists}$ is $\mathrm{NSOP}_{1}$. By Theorem 9.1 of
[12], it suffices to find an invariant ternary relation
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}$
between subsets of $\mathbb{M}$ over models with the following properties:

(a) Strong finite character: For all $a,b$ and $M\models T$, if
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}b$, then there is some formula
$\varphi(x,b)\in\mathrm{tp}(a/Mb)$, where $\varphi(x,y)$ has parameters in
$M$, such that for every $a^{\prime}\models\varphi(x,b)$,
$a^{\prime}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}b$.

(b) Existence over models: for all $a$ and $M\models T$,
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}M$.

(c) Monotonicity: For all $A^{\prime}\subseteq A$, $B^{\prime}\subseteq B$,
$M\models T$, $A\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}B$ implies
$A^{\prime}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}B^{\prime}$.

(d) Symmetry: For all $a,b$ and $M\models T$, $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}b$ implies
$b\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}a$ (and vice versa).

(e) The independence theorem: for any $a,a^{\prime},b,c$ and $M\models T$,
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}b$, $a^{\prime}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}c$,
$b\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}c$ and $a\equiv_{M}a^{\prime}$ implies that
there is $a^{\prime\prime}$ with $a^{\prime\prime}b\equiv_{M}ab$,
$a^{\prime\prime}c\equiv_{M}a^{\prime}c$ and
$a^{\prime\prime}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}bc$.

First, note that for any $A$, there is $A^{\prime}\supseteq A$ such that
$A^{\prime}\leq\mathbb{M}$ and for all $A^{\prime\prime}\supseteq A$ with
$A^{\prime\prime}\leq\mathbb{M}$, $A^{\prime}\subseteq A^{\prime\prime}$.
Denote this $\mathrm{cl}(A)$; clearly, this is contained in $\mathrm{acl}(A)$
(and it can even be checked that $\mathrm{cl}(A)=\mathrm{acl}(A)$). This
allows us to define $\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}$: $A\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}B$ if
$\mathrm{cl}(MA)\cap\mathrm{cl}(MB)=M$, and every ($R_{1}(\mathbb{M})\cup
R_{2}(\mathbb{M})$)-path between a point of $O(\mathrm{cl}(MA)\backslash M)$
and $O(\mathrm{cl}(MB)\backslash M)$ contains a point of $M$.

We see this has strong finite character: for $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}b$, $a_{0}\in\mathrm{cl}(aM)$,
$b_{0}\in\mathrm{cl}(bM)$ the endpoints of the path of length $n$ witnessing
this, let $\varphi_{1}(y,b)$ isolate $b_{0}$ over $Mb$, $\varphi_{2}(x)$ say
that $x$ is not within distance $n$ of the closest point of $M$ to some (any)
$b^{\prime}_{0}\models\varphi_{1}(x,b)$ (which would be required, should there
be a (non-self-overlapping) path of length $n$ between $x$ and $b_{0}$ going
through $M$), and $\varphi_{3}(x,b)$ say that there is a path of length $n$
from $x$ to some point $c$ satisfying $\models\varphi_{1}(c,b)$. Then
$\varphi(x,b)=:\varphi_{2}(x)\wedge\varphi_{3}(x,b)$ suffices.

Existence over models, monotonicity, and symmetry are immediate. So it remains
to show the independence theorem. By definition of $\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}$, we may assume
$a=\mathrm{cl}(aM)$ and similar for $a^{\prime},b,c$. We first give an
analysis of the structure of pairs $a,b$ with $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}b$:

######  Lemma 2.6.

Let $a=\mathrm{cl}(aM)$, $b=\mathrm{cl}(bM)$, and $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}b$. Then
$\mathrm{cl}(ab)=a^{*}b^{*}$, where for $P_{0}=P(ab)$, $a^{*}$ is the closure
of $a$ under the functions $p^{j}_{\to^{i}}$ for $p\in P_{0}$ (so in
particular, $P(a^{*})=P(a)$), and $b^{*}$ is the closure of $b$ under these
same functions. Every point of $O(a^{*}\backslash a)$ is connected by some
($R_{1}\vee R_{2}$)-path to a point of $a\backslash M$, and $a\leq a^{*}$;
similarly for $b^{*}$.

###### Proof.

The second sentence follows from the definition of the $p^{j}_{\to^{i}}$, and
the fact that $M,a,b\leq\mathbb{M}$; it remains to show that
$a^{*}b^{*}\leq\mathbb{M}$. But $a^{*}b^{*}$ is the closure of $ab$ under the
functions $p^{j}_{\to^{i}}$ for $p\in P_{0}$, so is complete, and condition
(i) is satisfied. Moreover, $O(a)O(b)\leq\mathbb{M}$ (i.e., condition (ii) of
being closed in $\mathbb{M}$ holds for $ab$): any path with endpoints in
$O(a)O(b)$ and intermediate points all in $O(\mathbb{M}\backslash ab)$ must,
because $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}b$ and $M\subseteq a,b$, have both
endpoints in either of $O(a)$ or $O(b)$, a contradiction because $O(a)$ and
$O(b)$ are closed in $\mathbb{M}$. Because every point of $O(a^{*})O(b^{*})$
is connected to $ab$ by the second clause of this lemma, condition (ii) of
$a^{*}b^{*}\leq\mathbb{M}$ then follows from $O(a)O(b)\leq\mathbb{M}$. ∎

In the following, for $P_{0}\subset P(\mathbb{M})$ and $A\subset\mathbb{M}$,
let $\mathrm{cl}_{P_{0}}(A)$ denote the closure of $A$ under $p^{j}_{\to^{i}}$
for $p\in P_{0}$, as in Lemma 2.6. We want to find
$\tilde{a},\tilde{b},\tilde{c}$ so that $\tilde{a}\tilde{b}\equiv_{M}ab$,
$\tilde{a}\tilde{c}\equiv_{M}a^{\prime}c$, $\tilde{b}\tilde{c}\equiv_{M}bc$,
$\tilde{a}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}\tilde{b}\tilde{c}$. Let
$b^{*_{c}}=\mathrm{cl}_{P(b)P(c)}(b)$, $c^{*_{b}}=\mathrm{cl}_{P(b)P(c)}(c)$,
$b^{*_{a}}=\mathrm{cl}_{P(a)P(b)}(b)$,
$c^{*_{a^{\prime}}}=\mathrm{cl}_{P(a^{\prime})P(c)}(c)$,
$a^{*_{b}}=\mathrm{cl}_{P(a)P(b)}(a)$,
$a^{*_{c}}=\mathrm{cl}_{P(a^{\prime})P(c)}(a^{\prime})$. We now build a
copacetic $\mathcal{L}$-structure extending $M$ as follows. Let
$A_{0}=:\tilde{a}_{0}\tilde{b}_{0}\tilde{c}_{0}$ where $R_{1},R_{2}$ is
defined on $O(\tilde{a}_{0}),O(\tilde{b}_{0}),O(\tilde{c}_{0})$ so that these
are freely amalgamated sets isomorphic to $O(a),O(b),O(c)$ over $O(M)$ (so in
particular,
$O(a)O(b)\equiv^{\mathcal{L}-\mathrm{qftp}}_{O(M)}O(\tilde{a}_{0})O(\tilde{b}_{0})$
because $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}b$, and so on), and so that
$P(\tilde{a}_{0}),P(\tilde{b}_{0}),P(\tilde{c}_{0})$ and $\rho_{1},\rho_{2}$
are defined so that
$\tilde{a}_{0}\tilde{b}_{0}\equiv^{\mathcal{L}-\mathrm{qftp}}_{M}ab$,
$\tilde{a}_{0}\tilde{c}_{0}\equiv^{\mathcal{L}-\mathrm{qftp}}_{M}a^{\prime}c$,
$\tilde{b}_{0}\tilde{c}_{0}\equiv^{\mathcal{L}-\mathrm{qftp}}_{M}bc$; this
makes sense because $a\equiv_{M}a^{\prime}$. Then $A_{0}$ is copacetic,
because the only way axiom (C4) can fail is, without loss of generality, when
there is $p\in P(\tilde{a}_{0}\backslash M)$, $o_{1}\in
O(\tilde{b}_{0}\backslash M)$, $o_{2}\in O(\tilde{c}_{0}\backslash M)$ so that
$o_{1},o_{2}$ lie on the boundary of some common $R_{1}$-ball and
$A_{0}\models\rho_{1}(p,o_{1})\wedge\rho_{1}(p,o_{2})$. Because
$O(\tilde{b}_{0}),O(\tilde{c}_{0})$ are freely amalgamated over $M$, the only
way the former can happen is for $o_{1}$ and $o_{2}$ to have a common
$R_{1}$-neighbor $m\in M$. But then (say)
$o_{1}=p^{1}_{\to^{1}}(m)\in\tilde{a}_{0}$, contradicting
$\tilde{a}_{0}\cap\tilde{b}_{0}=M$. Now extend $A_{0}$ to
$A_{1}=:\tilde{a}^{*_{\tilde{b}}}\tilde{a}^{*_{\tilde{c}}}\tilde{b}^{*_{\tilde{a}}}\tilde{b}^{*_{\tilde{c}}}\tilde{c}^{*_{\tilde{a}}}\tilde{c}^{*_{\tilde{b}}}$,
where

(1)$\tilde{a}^{*_{\tilde{b}}}=\tilde{a}_{0}\cup
O(\tilde{a}^{*_{\tilde{b}}}\backslash\tilde{a}_{0})$,
$\tilde{a}^{*_{\tilde{c}}}=\tilde{a}_{0}\cup
O(\tilde{a}^{*_{\tilde{c}}}\backslash\tilde{a}_{0})$,
$\tilde{b}^{*_{\tilde{a}}}=\tilde{b}_{0}\cup
O(\tilde{b}^{*_{\tilde{a}}}\backslash\tilde{b}_{0})$,
$\tilde{b}^{*_{\tilde{c}}}=\tilde{b}_{0}\cup
O(\tilde{b}^{*_{\tilde{c}}}\backslash\tilde{b}_{0})$,
$\tilde{c}^{*_{\tilde{a}}}=\tilde{c}_{0}\cup
O(\tilde{c}^{*_{\tilde{a}}}\backslash\tilde{c}_{0})$,
$\tilde{c}^{*_{\tilde{b}}}=\tilde{c}_{0}\cup
O(\tilde{c}^{*_{\tilde{b}}}\backslash\tilde{c}_{0})$,

(2) $O(\tilde{a}^{*_{\tilde{b}}}\backslash\tilde{a}_{0})$,
$O(\tilde{a}^{*_{\tilde{c}}}\backslash\tilde{a}_{0})$,
$O(\tilde{b}^{*_{\tilde{a}}}\backslash\tilde{b}_{0})$,
$O(\tilde{b}^{*_{\tilde{c}}}\backslash\tilde{b}_{0})$,
$O(\tilde{c}^{*_{\tilde{a}}}\backslash\tilde{c}_{0})$,
$O(\tilde{c}^{*_{\tilde{b}}}\backslash\tilde{c}_{0})$ are pairwise disjoint
and disjoint from $A_{0}$

(3) The only $R_{i}$-edges of $A_{1}$ are those of $A_{0}$, as well as those
required to make
$O(\tilde{a}^{*_{\tilde{b}}})\equiv^{\mathcal{L}-\mathrm{qftp}}_{O(M)}O(a^{*_{b}})$,
$O(\tilde{a}^{*_{\tilde{c}}})\equiv^{\mathcal{L}-\mathrm{qftp}}_{O(M)}O(a^{\prime*_{c}})$,
$O(\tilde{b}^{*_{\tilde{a}}})\equiv^{\mathcal{L}-\mathrm{qftp}}_{O(M)}O(b^{*_{a}})$,
$O(\tilde{b}^{*_{\tilde{c}}})\equiv^{\mathcal{L}-\mathrm{qftp}}_{O(M)}O(b^{*_{c}})$,
$O(\tilde{c}^{*_{\tilde{a}}})\equiv^{\mathcal{L}-\mathrm{qftp}}_{O(M)}O(c^{*_{a^{\prime}}})$,
$O(\tilde{c}^{*_{\tilde{b}}})\equiv^{\mathcal{L}-\mathrm{qftp}}_{O(M)}O(c^{*_{b}})$.

(4) Let us extend $\rho_{1}\rho_{2}$ where required so that
$\tilde{a}^{*_{\tilde{b}}}\tilde{b}^{*_{\tilde{a}}}\equiv^{\mathcal{L}-\mathrm{qftp}}_{M}a^{*_{b}}b^{*_{a}}$,
$\tilde{a}^{*_{\tilde{c}}}\tilde{c}^{*_{\tilde{a}}}\equiv^{\mathcal{L}-\mathrm{qftp}}_{M}a^{\prime*_{c}}c^{*_{a^{\prime}}}$,
$\tilde{b}^{*_{\tilde{c}}}\tilde{c}^{*_{\tilde{b}}}\equiv^{\mathcal{L}-\mathrm{qftp}}_{M}b^{*_{c}}c^{*_{b}}$.
Because
$\tilde{a}^{*_{\tilde{b}}}\tilde{b}^{*_{\tilde{a}}},\tilde{a}^{*_{\tilde{c}}}\tilde{c}^{*_{\tilde{a}}},\tilde{b}^{*_{\tilde{c}}}\tilde{c}^{*_{\tilde{b}}}$
are already known to satisfy (C4), a failure of (C4), which would then be
witnessed by $p\in P(A_{1})$, $o_{1},o_{2},o_{3}\in O(A_{1})$ (with the last
two perhaps equal), could now happen in the following two cases (among the
instances where the $\rho_{i}$ are yet defined), both of which we rule out.
First, one of the $o_{i}$ is in $a^{\dagger}\backslash{A_{0}}$, where
$a^{\dagger}$ is one of
$\tilde{a}^{*_{\tilde{b}}}\tilde{a}^{*_{\tilde{c}}},\tilde{b}^{*_{\tilde{a}}}\tilde{b}^{*_{\tilde{c}}},\tilde{c}^{*_{\tilde{a}}}\tilde{c}^{*_{\tilde{b}}}$,
and the other is in $A_{1}\backslash a^{\dagger}$. But then, by the
connectedness claim in the second clause of Lemma 2.6, (3) and the
construction of $O(A_{0})$ tell us that these two $o_{i}$ must have distance
at least $2$ apart, so a failure of (C4) cannot arise here. The other case is
where $o_{1},o_{2},o_{3}\in a^{\dagger}$ for $a^{\dagger}$ one of
$\tilde{a}^{*_{\tilde{b}}}\tilde{a}^{*_{\tilde{c}}},\tilde{b}^{*_{\tilde{a}}}\tilde{b}^{*_{\tilde{c}}},\tilde{c}^{*_{\tilde{a}}}\tilde{c}^{*_{\tilde{b}}}$,
and some two do not belong to the same
$\tilde{a}^{*_{\tilde{b}}},\tilde{a}^{*_{\tilde{c}}},\tilde{b}^{*_{\tilde{a}}},\tilde{b}^{*_{\tilde{c}}},\tilde{c}^{*_{\tilde{a}}},\tilde{c}^{*_{\tilde{b}}}$;
then $p\in a^{\dagger}$ because this is the only way the $\rho_{i}$ can be
defined so far for $p$, $o_{1},o_{2},o_{3}$. Then these two $o_{j}$, say
$o_{1}$ and $o_{2}$ must satisfy $\rho_{i}(p,o_{1}),\rho_{i}(p,o_{2})$ for
some $i\in\\{1,2\\}$, and must be common $R_{i}$-neighbors of the evident
$\tilde{a},\tilde{b},\tilde{c}$ while lying outside of this
$\tilde{a},\tilde{b},\tilde{c}$. But this is impossible, by the claim $a\leq
a^{*}$, $b\leq b^{*}$ of Lemma 2.6. Therefore, we do not yet get a failure of
(C4), and it remains to extend $\rho_{1},\rho_{2}$ to get (C3) while
maintaining (C4), which we do in the next step.

(5) By the claim $a\leq a^{*}$, $b\leq b^{*}$ in the second clause of Lemma
2.6, we can use Claim 2.1 to extend $\rho_{1},\rho_{2}$ where not yet defined,
maintaining (C4) by the description of the $R_{1},R_{2}$-structure in (3), and
thereby producing a copacetic $\mathcal{L}$-structure.

Observe also the following:

(6) In $A_{0}$, there is no path between $\tilde{a}$ and $\tilde{b}\tilde{c}$
not going through $M$. So by the connectedness claim in Lemma 2.6 applied to
$c^{*_{b}},b^{*_{c}}$, in $A_{1}$ there is no path between $\tilde{a}$ and
$\tilde{c}^{*_{\tilde{b}}}\tilde{b}^{*_{\tilde{c}}}$ not going through $M$.

(7). By construction, $O(\tilde{a}),O(\tilde{b}),O(\tilde{c})$ are closed in
$O(A_{1})$. So by the proof of the first clause of Lemma 2.6,
$\tilde{a}^{*_{\tilde{b}}}\tilde{b}^{*_{\tilde{a}}},\tilde{a}^{*_{\tilde{c}}}\tilde{c}^{*_{\tilde{a}}},\tilde{b}^{*_{\tilde{c}}}\tilde{c}^{*_{\tilde{b}}}$
are each closed in $A_{1}$ (and are also each complete).

Finally, we extend $A_{1}$ to a complete copacetic $\mathcal{L}$-structure, so
that (6) and (7) still hold replacing $A_{1}$ with $A_{2}$; since
$\tilde{a}^{*_{\tilde{b}}}\tilde{b}^{*_{\tilde{a}}},\tilde{a}^{*_{\tilde{c}}}\tilde{c}^{*_{\tilde{a}}},\tilde{b}^{*_{\tilde{c}}}\tilde{c}^{*_{\tilde{b}}}$
are complete, to preserve (7) we must just preserve clause (ii). We can extend
$A_{1}$ to a complete copacetic $\mathcal{L}$-structure just by repeated
applications of the proof of (ii). But notice that this proceeds just by
successively adding nodes in the sort $O$ with exactly one $R_{1}\vee
R_{2}$-neighbor in the previously added nodes, so adds no new paths between
nodes in $O(A_{1})$. So (6) and (7) are in fact preserved.

Note that $A\leq B\leq C$ implies $A\leq C$, so by (4) and (7) (i.e., the
version where $A_{1}$ is replaced with $A_{2}$), $M\leq A_{2}$. Now use Lemma
2.4 to obtain an embedding $\iota:A_{2}\hookrightarrow\mathbb{M}$ which is the
identity on $M$, and such that $\iota(A_{2})\leq\mathbb{M}$. Let
$\tilde{a}=\iota(\tilde{a}_{0})$, $\tilde{b}=\iota(\tilde{b}_{0})$,
$\tilde{c}=\iota(\tilde{c}_{0})$. Then (using Corollary 2.5) by (4), (7)
(again, the version where $A_{1}$ is replaced with $A_{2}$), $\iota(A_{2})\leq
M$, and the first clause of Lemma 2.6, $\tilde{a}\tilde{b}\equiv_{M}ab$,
$\tilde{a}\tilde{c}\equiv_{M}a^{\prime}c$, $\tilde{b}\tilde{c}\equiv_{M}bc$.
Moreover, by (6) (yet again, the version where $A_{1}$ is replaced with
$A_{2}$) and $\iota(A_{2})\leq M$ , $\tilde{a}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}\tilde{b}\tilde{c}$. So we have proven the
independence theorem for $\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}$, and $T^{\not\exists}$ is
$\mathrm{NSOP}_{1}$.

It remains to show that $T^{\not\exists}$ does not satisfy the existence
axiom. Let $p(x)$ be the unique type in the sort $P$ (in one variable) over
$\emptyset$. Let $o\in O(\mathbb{M})$. Then
$p(x)\vdash\rho_{1}(x,o)\vee\rho_{2}(x,o)$, by (C3). We show that
$\rho_{1}(x,o)$ $2$-divides over $\emptyset$. We can find an
$\emptyset$-indiscernible sequence $\\{o_{i}\\}_{i<\omega}$, $o_{0}=o$, so
that the $o_{i}$ lie on the boundary of some fixed $R_{1}$-ball of radius $1$.
Then $\\{\rho(x,o_{i})\\}_{i<\omega}$ is $2$-inconsistent by (C4), so
$\rho_{1}(x,o_{i})$ $2$-divides over $\emptyset$. That $\rho_{2}(x,o_{i})$
$3$-divides over $\emptyset$ will be similar. So $p\in S(\emptyset)$ forks
over $\emptyset$, violating the existence axiom.

This proves the main theorem of this paper, Theorem 1.5, and answers the main
question, Question 1.4.

######  Remark 2.7.

It is not too hard to show that $\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}$ above even satisfies the following axiom:

(f) Witnessing: Let $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}b$, and let $\\{b_{i}\\}_{i<\omega}$,
$b_{0}=b$, be an $M$-indiscernible sequence with
$b_{i}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{M}b_{0}\ldots b_{i-1}$ for all $i<\omega$.
Then there is a formula $\varphi(x,b)\in\mathrm{tp}(a/Mb)$, $\varphi(x,y)\in
L(M)$, so that $\\{\varphi(x,b_{i})\\}_{i<\omega}$ is inconsistent.

Theorem 6.11 of [13] says that if an invariant ternary relation
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}$
between subsets of $\mathbb{M}$ over models satisfies strong finite character,
existence over models, monotonicity, symmetry, the independence theorem and
witnessing, then $\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}$ coincides with Kim-independence
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}$ (Definition 1.1.) So in
$T^{\not\exists}$, Kim-independence (over models) is given by
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}$. Because $T^{\not\exists}$ does not satisfy
the existence axiom, the results on Kim-independence over sets from [9], [2])
do not apply to $T^{\not\exists}$. But note that it makes sense to define
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}b$ the same way as above when
($\mathrm{acl}(C)=$) $\mathrm{cl}(C)=C$, and, when $C$ is any set, define
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}b$ by $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{\mathrm{cl}(C)}b$, giving a ternary relation
on sets. Over sets, by the same proofs as above, $\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}$ satisfies the
analogues of strong finite character, monotonicity, symmetry, the independence
theorem (where $\equiv_{C}$ is replaces by $\equiv^{\mathrm{Lstp}}_{C}$,
though this is the same as $\equiv_{\mathrm{cl}(C)}$, with respect to which
the independence theorem holds for $\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}$), and witnessing; moreover
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}$
satisfies a stronger version of existence over sets:

(b′) Existence and extension over sets: for any $a,C$ and $B^{\prime}\supseteq
B\supseteq C$, $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}C$, and if $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{C}B$ there is
$a^{\prime}\equiv_{B}a$ with $a^{\prime}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{C}B^{\prime}$.

Ramsey, in a presentation at the Banff International Research Station on joint
work with Itay Kaplan ([22]), defines the assertion that Kim-independence is
defined over sets to mean that there is a ternary relation
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}$
between sets satisfying the analogues over sets of strong finite character,
monotonicity, symmetry, the independence theorem, and witnessing, as well as
existence and extension over sets, and shows that such
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}$
is uniquely determined when it exists. So by [9], [2], in any
$\mathrm{NSOP}_{1}$ theory satisfying the existence axiom, Kim-independence is
defined over sets in the sense of [22]. But also, despite $T^{\not\exists}$
not satisfying the existence axiom, Kim-independence is defined over sets in
this sense in the theory $T^{\not\exists}$, even if the results of [9] on Kim-
independence as defined by Dobrowolski, Kim, and Ramsey (Definition 1.3 above)
do not apply in $T^{\not\exists}$.222In fact, in $T^{\not\exists}$,
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}$
actually coincides with $\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}$ as defined by [9] (i.e. Kim-forking
independence with respect to nonforking Morley sequences, Definition 1.3
above), so the conclusions of, say, Corollary 4.9 or Theorem 5.6 of [9] hold:
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}$ as defined there is symmetric, and
satisfies the independence theorem, over arbitrary sets. (If
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}b$ and $aC$ is algebraically closed, then
$\mathrm{tp}(a/Cb)$ implies a finite disjunction of formulas of the form
$\varphi(x,b^{\prime})$, where $b^{\prime}\notin\mathrm{cl}(C)$ is a singleton
of $O$ or $P$ and $\varphi(x,b^{\prime})$ either says that $x=b^{\prime}$ or
implies that there is a path between $x$ and $b^{\prime}$ with no points in
$\mathrm{cl}(C)$, and a formula of either kind divides over $C$ with respect
to a $\mathrm{cl}(C)$-invariant Morley sequence. On the other hand,
$a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}_{C}b$ implies $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{C}M$ for $M$
some $|C|^{+}$-saturated model containing $Cb$, so $a\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}^{K}_{C}M$ by
the independence theorem and $|C|^{+}$-saturatedness of $M$; see the clause
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}\Rightarrow\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}^{K}$ of Theorem
9.1 of [12], and the standard argument that forking-dependence on a
sufficiently saturated model implies dividing-dependence on that model.) But
Proposition 4.9 of [9] fails–it is not necessarily true that $\varphi(x,b)$
forks over $C$ with respect to nonforking Morley sequences if and only if it
divides with respect to nonforking Morley sequences. For example, for $o\in
O,p\in P$, let $\varphi(x,op)=:x=p$; then $\varphi(x,op)$ does not divide with
respect to a nonforking Morley sequence over $\emptyset$ (i.e. Kim-divide over
$\emptyset$, as in Definition 1.3), because there are no nonforking Morley
sequences over $\emptyset$ starting with $op$), but it implies
$\tilde{\varphi}(x,p)=:x=p$, which does divide with respect to a nonforking
Morley sequence over $\emptyset$, so $\varphi(x,op)$ Kim-forks over
$\emptyset$. Moreover, Kim’s lemma, Theorem 3.5 of [9], is also false in
$T^{\not\exists}$: $\varphi(x,op)$ divides over $\emptyset$ with respect to
all nonforking Morley sequences over $\emptyset$ starting with $op$, but not
with respect to some nonforking Morley sequence over $\emptyset$ starting with
$op$! By way of obtaining an $\mathrm{NSOP}_{1}$ theory where
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}$ as defined over sets by [9] (Defintion
1.3 here) does not, say, satisfy the independence theorem, we expect that, by
an extremely tedious verification, $T^{\not\exists}$ can be shown to eliminate
$\exists^{\infty}$. So by Theorem 5 of [25] and Theorem 4.5 of [18], the
generic expansion of $T^{\not\exists}$ by functions from $P$ to $O$ and from
$O$ to $p$ (i.e. the model companion of models of (the Morleyization of)
$T^{\not\exists}$ expanded by a unary function from sort $P$ to sort $O$ and a
unary function from sort $O$ to sort $P$) will exist and have
$\mathrm{NSOP}_{1}$, and no consistent formula can Kim-divide over
$\emptyset$, because every nonempty parameter will have an element of $O$ and
an element of $P$ in its definable closure, so can be shown to begin no Morley
sequence over $\emptyset$ as in the original proof that $T^{\not\exists}$ does
not satisfy the existence axiom. So, using Definition 1.3 to define
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}$ over arbitrary sets, any set will be Kim-
independent over $\emptyset$ from any nonempty set.  So the results stated in
[22] are independent of the previous work on the existence axiom.

##  3\. Quantitative results

Doborowolski, Kim, and Ramsey show, in Remark 6.7 of [9], that in a theory
without the strict order property (i.e. an $\mathrm{NSOP}$ theory), the
failure of the existence axiom cannot be witnessed by two formulas that
$2$-divide:

######  Fact 3.1.

Let $T$ be $\mathrm{NSOP}$, and $p\in S(A)$. Then there are no formulas
$\varphi_{1}(x,b_{1})$, $\varphi_{2}(x,b_{2})$, each of which $2$-divide over
$A$, such that $p\vdash\varphi_{1}(x,b_{1})\vee\varphi_{2}(x,b_{2})$.

In the previous section, we gave an example, $T^{\not\exists}$, of an
$\mathrm{NSOP}_{1}$ theory where, for $p\in S(\emptyset)$,
$p\vdash\varphi_{1}(x,b)\vee\varphi_{2}(x,b)$, where $\varphi_{1}(x,b)$
$2$-divides over $\emptyset$ and $\varphi_{2}(x,b)$ $3$-divides over
$\emptyset$. Here, we describe an example, $T^{\not\exists^{2,2,2}}$, of an
$\mathrm{NSOP}_{1}$ theory where, for $p\in S(\emptyset)$,
$p\vdash\varphi_{1}(x,b)\vee\varphi_{2}(x,b)\vee\varphi_{3}(x,b)$, where for
$i=1,2,3$ each $\varphi_{i}(x,b)$ $2$-divides over $\emptyset$. This will show
the optimality of Fact 3.1.

Let $\mathcal{L}$ be the language with sorts $P$ and $O$, symbols $R_{1}$,
$R_{2}$ and $R_{3}$ for binary relations on $O$, and symbols $\rho_{1}$,
$\rho_{2}$, and $\rho_{3}$ for binary relations between $P$ and $O$. Call an
$\mathcal{L}$-structure $A$ copacetic2,2,2 if:

(C1)2,2,2 For $i=1,2,3$, $R_{i}(A)$ is a symmetric, irreflexive relation on
$O(A)$, and the three are mutually exclusive: for $a_{1},a_{2}\in O(A)$,
$A\not\models R_{i}(a_{1},a_{2})\wedge R_{j}(a_{1},a_{2})$ for $i\neq
j\in\\{1,2,3\\}$

(C2)2,2,2 The relation $R_{1}(A)\cup R_{2}(A)\cup R_{3}(A)$ has no loops on
$O(A)$ (i.e. there are no distinct $a_{0}\ldots a_{n-1}\in O(A)$, $n>2$, and
$i_{1}\ldots i_{n}\in\\{1,2,3\\}$ so that, for $0\leq j\leq n-1$, $A\models
R_{i_{j}}(a_{i},a_{i+1\mathrm{\>mod\>}n})$).

(C3)2,2,2 For all $b\in P(A)$, $a\in O(A)$, exactly one of
$A\models\rho_{1}(b,a)$, $A\models\rho_{2}(b,a)$, and $\rho_{3}(b,a)$ hold.

(C4)2,2,2: For $i\in\\{1,2,3\\}$, there is no $b\in P(A)$ and distinct
$a_{1},a_{2}$ on the boundary of some fixed unit $R_{i}$-ball so that
$A\models\rho_{i}(b,a_{1})\wedge\rho_{i}(b,a_{2})$.

We define the closure relation $\leq$ analogously to the previous section, and
construct a theory satisfying the analogous statement to Lemma 2.4, which will
be $\mathrm{NSOP}_{1}$ and satisfy
$p\vdash\rho_{1}(x,o)\vee\rho_{2}(x,o)\vee\rho_{3}(x,o)$ for any $o\in
O(\mathbb{M})$ and $p\in S(\emptyset)$ the unique type (in one variable) in
sort $P$ over $\emptyset$; $\rho_{i}(x,o)$ will $2$-divide over $\emptyset$
for $i\in\\{1,2,3\\}$, as desired. The entire proof is a straightforward
generalization of the previous section, with a single exception: in place of
Subclaim 2.3, we must prove the below subclaim. Let $O$ be an undirected graph
without cycles and with a $3$-coloring of its edges, with $R_{1}$, $R_{2}$,
$R_{3}$ denoting edges of each color. Let $\rho_{1},\rho_{2},\rho_{3}\subset
O$, $O=\rho_{1}\cup\rho_{2}\cup\rho_{3}$, $\rho_{i}\cap\rho_{j}=\emptyset$ for
$i\neq j\in\\{1,2,3\\}$ be a coloring of the vertices of $O$ so that, for
$i=\\{1,2,3\\}$, no two distinct vertices of $O$, lying on the boundary of the
same $R_{i}$-ball of radius $1$ (i.e. they have a common $R_{i}$-neighbor),
are both colored by $\rho_{i}$. Then we call $\rho_{1},\rho_{2},\rho_{3}$ a
(C4)2,2,2-coloring of $O$.

######  Subclaim 3.2.

Let $O$ be a connected graph without cycles, and with a $3$-coloring of its
edges. Let $O_{1},O_{2}\subset O$ be connected subgraphs so that each vertex
of $O_{1}$ has distance at least $5$ from each vertex of $O_{2}$. For $i=1,2$,
let $\rho_{1}^{i},\rho_{2}^{i},\rho_{3}^{i}$ be a (C4)2,2,2-coloring of
$O_{i}$. Then there is a (C4)2,2,2-coloring $\rho_{1},\rho_{2},\rho_{3}$ of
some connected set $O^{\prime}$ containing $O_{1}$ and $O_{2}$, where for
$i=1,2$, $\rho_{1},\rho_{2},\rho_{3}$ extends
$\rho_{1}^{i},\rho_{2}^{i},\rho_{3}^{i}$ on $O_{i}$.

###### Proof.

As in the proof of Subclaim 2.3, let $O^{\prime}=O_{1}\cup O_{2}\cup I$ where
$I$ is the shortest path between $O_{1}$ and $O_{2}$, and let $I$ consist,
ordered in the direction from $O_{1}$ to $O_{2}$, of $o_{0},\ldots o_{n}$, for
$o_{0}\in O_{1}$, $o_{n}\in O_{2}$, $o_{1}\ldots o_{n-1}\in
O\backslash(O_{1}\cup O_{2})$, and $n\geq 5$. Again, as in that proof, color
$O_{i}$ by $\rho^{i}_{1},\rho^{i}_{2},\rho^{i}_{3}$ for $i\in\\{1,2\\}$, color
$o_{1}$ by $\rho_{i}$ where $i$ is such that $o_{1}$ is not an
$R_{i}$-neighbor of $o_{0}$, and color $o_{n-1}$ by $\rho_{j}$ where $j$ is
such that $o_{n-1}$ is not an $R_{j}$-neighbor of $o_{n}$–then as before, the
condition of being a (C4)2,2,2-coloring cannot fail at the boundary of a unit
ball centered at a point of $O_{1}$ or $O_{2}$. Now let
$n_{\mathrm{even}},n_{\mathrm{odd}}$, respectively, be the least even and odd
numbers less than $n-1$. Then, because there are three colors available, we
can color $o_{2},\ldots,o_{2i},\ldots,o_{n_{\mathrm{even}}}$ so that each
vertex in the sequence
$o_{0},o_{2},\ldots,o_{2i},\ldots,o_{n_{\mathrm{even}}},o_{n_{\mathrm{even}}+2}$
is colored differently from the previous vertex in that sequence–noting that
the colors of $o_{0}$ and $o_{n_{\mathrm{even}}+2}$ are already decided,
alternate the color of $o_{0}$ with a color distinct from that of $o_{0}$ and
$o_{n_{\mathrm{even}}+2}$. Similarly, we can color
$o_{3},\ldots,o_{2i+1},\ldots o_{n_{\mathrm{odd}}}$ so that each vertex in the
sequence
$o_{1},o_{3},\ldots,o_{2i+1},\ldots,o_{n_{\mathrm{odd}}},o_{n_{\mathrm{odd}}+2}$
is colored differently from the previous vertex in that sequence. Coloring the
intermediate vertices $o_{2},\ldots o_{n-2}$ according to these observations,
we see that the condition of being a (C4)2,2,2-coloring cannot fail on the
boundary of a unit ball centered at one of $o_{1},\ldots o_{n-1}$, because the
boundary of such a ball will always be colored by two different colors. ∎

Note that a similar subclaim would fail, were we to try to use an analogous
construction to obtain an $\mathrm{NSOP}_{1}$ theory where, for $p\in
S(\emptyset)$, $p\vdash\varphi_{1}(x,b_{1})\vee\varphi_{2}(x,b_{2})$ for
$\varphi_{i}(x,b_{i})$ $2$-dividing over $\emptyset$.

##  4\. Open questions

The theory $T^{\not\exists}$, despite being an $\mathrm{NSOP}_{1}$ theory that
does not satisfy the existence axiom, is not countably categorical. Motivated
by this, we ask:

######  Question 4.1.

Does every countably categorical $\mathrm{NSOP}_{1}$ (or even $\mathrm{NSOP}$)
theory satisfy the existence axiom?

Moreover, in $T^{\not\exists}$, Kim-independence over models is not just given
by the operation $\mathrm{acl}^{eq}$; see Remark 2.7. In Definition 6.10 of
[11], the definition of the property of being one-based is extended (up to
elimination of hyperimaginaries) from simple theories to $\mathrm{NSOP}_{1}$
theories:

######  Definition 4.2.

Let $T$ be an $\mathrm{NSOP}_{1}$ theory. Then $T$ is _one-based_ if
$A\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar
12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern
5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox
to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern
5.71527pt\scriptscriptstyle{}}}^{K}_{M}B$ implies (equivalently, is equivalent
to) $\mathrm{acl}^{eq}(AM)\cap\mathrm{acl}^{eq}(BM)\supsetneq M$.

So $T^{\not\exists}$ is not one-based. (See Example 4.6.1 of [20].) This leads
us to ask:

######  Question 4.3.

Does every one-based $\mathrm{NSOP}_{1}$ theory satisfy the existence axiom?

Recall that, as stated in Remark 2.7, Kim-independence is defined over sets in
any $\mathrm{NSOP}_{1}$ theory satisfying the existence axiom, but is also
defined over sets in $T^{\not\exists}$ despite $T^{\not\exists}$ not
satisfying the existence axiom. A final question, motivated by this remark and
by the original motivation discussed in the introduction for Question 1.4, the
main question of this paper, is asked by Ramsey:

######  Question 4.4.

([22], [21]) Is Kim-independence defined over sets in every
$\mathrm{NSOP}_{1}$ theory?

## Acknowledgements

The author would like to thank James Freitag, Maryanthe Malliaris and Nicholas
Ramsey for many insightful conversations. In particular, conversations with
Nicholas Ramsey were instrumental in inspiring the discussion in Remark 2.7
and Question 4.4 of this paper.

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