1 Introduction
This paper furthers the development of the theory of Kimindependence in the context of NSOP $_{1}$ theories satisfying the existence axiom. Building on earlier work in [Reference Chernikov and Ramsey4] and a suggestion of the secondnamed author [Reference Kim11], Kimindependence was introduced in [Reference Kaplan and Ramsey7], where it was shown to be a wellbehaved notion of independence in NSOP $_{1}$ theories. This work established a strong analogy between the theory of nonforking independence in simple theories and Kimindependence in NSOP $_{1}$ theories, an analogy which subsequent works have only deepened [Reference Kaplan and Ramsey8, Reference Kaplan, Ramsey and Shelah9, Reference Kim13]. Nonetheless, one major difference between the two notions of independence is that, unlike nonforking which makes sense over all sets, Kimindependence is only a sensible notion of independence over models: Kimindependence is defined in terms of formulas that divide with respect to a Morley sequence in a global invariant type, and such a sequence, in general, is only guaranteed to exist over a model. In [Reference Dobrowolski, Kim and Ramsey5], the second and thirdnamed author, together with Dobrowolski, focused on the context of NSOP $_{1}$ theories that satisfy the existence axiom. There, it was shown that Kimindependence may be defined over arbitrary sets and basic theorems of Kimindependence over models hold in this broader context.
The existence axiom states that every complete type has a global nonforking extension, i.e., every set is an extension base in the terminology of [Reference Chernikov and Kaplan3]. This is equivalent to the statement that, in every type, there is a (nonforking) Morley sequence and, hence, assuming existence, one may redefine Kimindependence in terms of the formulas that divide along Morley sequences of this kind. New technical challenges arise in this setting, but in [Reference Dobrowolski, Kim and Ramsey5] it was shown that Kimindependence satisfies Kim’s lemma, symmetry, and the independence theorem for Lascar strong types. Moreover, all simple theories and all known examples in the growing list of NSOP $_{1}$ theories satisfy existence, and it is expected to hold in all NSOP $_{1}$ theories (see, e.g., [Reference Dobrowolski, Kim and Ramsey5, Fact 2.14]).
Here we continue work on Kimindependence in NSOP $_{1}$ theories satisfying existence, in particular, exploring aspects of the theory that are too cumbersome or uninteresting over models. In Section 3, we show that Kimindependence is transitive in an NSOP $_{1}$ theory satisfying existence and that, moreover, Kimdividing is witnessed by Morley sequences. These results were first established over models for all NSOP $_{1}$ theories in [Reference Kaplan and Ramsey8] and our proofs largely follow the same strategy. Nonetheless, suitable replacements need to be found for notions that only make sense, in general, over models, like heirs and coheirs. We find that arguments involving these notions can often be replaced by an argument involving a treeinduction, as in the construction of tree Morley sequences in [Reference Kaplan and Ramsey8]. In Section 4, we apply these results to low NSOP $_{1}$ theories satisfying existence, showing that Shelah strong types and Lascar strong types coincide, generalizing a result of Buechler for simple theories [Reference Buechler1] (see also [Reference Kim12, Reference Shami17]). In Section 5, we introduce a notion of rank for NSOP $_{1}$ theories and establish some of its basic properties. Finally, in Section 6, we generalize the Kim–Pillay criterion for Kimindependence from [Reference Chernikov and Ramsey4, Theorem 6.1] and [Reference Kaplan and Ramsey8, Theorem 6.11] to give a criterion for NSOP $_{1}$ in theories satisfying existence, which, additionally, gives an abstract characterization of Kimindependence over arbitrary sets in this setting.
2 Preliminaries
In this paper, T will always be a complete theory with monster model $\mathbb {M}$ . We will implicitly assume all models and sets of parameters are small, that is, of cardinality less than the degree of saturation and homogeneity of $\mathbb {M}$ . If we discuss an Iindexed indiscernible sequence $(a_{i})_{i \in I}$ , we will implicitly assume I is linearly ordered by $<$ and, given $i \in I$ , we will write $a_{<i}$ and $a_{\leq i}$ for the subsequences $(a_{j})_{j < i}$ and $(a_{j})_{j \leq i}$ respectively.
Definition 2.1. Suppose A is a set of parameters.

(1) We say that a formula $\varphi (x;a)$ divides over a set A if there is an Aindiscernible sequence $\langle a_{i} : i < \omega \rangle $ with $a_{0} = a$ such that $\{\varphi (x;a_{i}) : i < \omega \}$ is inconsistent.

(2) A formula $\varphi (x;a)$ is said to fork over A if $\varphi (x;a) \vdash \bigvee _{i < k} \psi _{i}(x;c_{i})$ , for some $k < \omega $ , with $\psi _{i}(x;c_{i})$ dividing over A.

(3) We say a partial type divides (forks) over A if it implies a formula that divides (forks) over A.

(4) For tuples a and b, we write or to indicate that $\text {tp}(a/Ab)$ does not divide over A or does not fork over A, respectively.

(5) A Morley sequence $(a_{i})_{i \in I}$ over A is an infinite Aindiscernible sequence such that for all $i \in I$ . If $p \in S(A)$ , we say $(a_{i})_{i \in I}$ is a Morley sequence in p if, additionally, $a_{i} \models p$ for all $i \in I$ .
The following is one of the key definitions of this paper. It defines a context in which Kimindependence may be studied over arbitrary sets.
Definition 2.2. We define the existence axiom to be any one of the following equivalent conditions on T:

(1) For all parameter sets A, any type $p \in S(A)$ does not fork over A.

(2) For all parameter sets A, no consistent formula over A forks over A.

(3) For all parameter sets A, every type $p \in S(A)$ has a global extension that does not fork over A.

(4) For all parameter sets A and any $p\in S(A)$ , there is a Morley sequence in p.
If T satisfies the existence axiom, we will often abbreviate this by writing T is with existence. See, e.g., [Reference Dobrowolski, Kim and Ramsey5, Remark 2.6] for the equivalence of (1)–(4).
Under existence, we may define Kimindependence over arbitrary sets. The following definition was given in [Reference Dobrowolski, Kim and Ramsey5], but it was observed already in [Reference Kaplan and Ramsey7, Theorem 7.7] that this agrees with the original definition over models.
Definition 2.3. Suppose T satisfies the existence axiom.

(1) We say a formula $\varphi (x;a)$ Kimdivides over A if there is a sequence $\langle a_{i} : i < \omega \rangle $ which is a Morley sequence over A with $a_{0} = a$ and $\{\varphi (x;a_{i}) : i < \omega \}$ inconsistent.

(2) A formula $\varphi (x;a)$ is said to Kimfork over A if $\varphi (x;a) \vdash \bigvee _{i < k} \psi _{i}(x;c_{i})$ , where each $\psi _{i}(x;c_{i})$ Kimdivides over A.

(3) We say a type Kimdivides (Kimforks) over A if it implies a formula that Kimdivides (Kimforks) over A.

(4) For tuples a and b, we write to indicate that $\text {tp}(a/Ab)$ does not Kimdivide over A.

(5) An Morley sequence $(a_{i})_{i \in I}$ over A is an infinite Aindiscernible sequence such that for all $i \in I$ .
Remark 2.4. By Kim’s lemma [Reference Kim12, Proposition 2.2.6], if T is simple, a formula Kimdivides over a set A if and only if it divides over A.
Definition 2.5 [Reference Džamonja and Shelah6, Definition 2.2].
The formula $\varphi (x;y)$ has SOP $_{1}$ if there is a collection of tuples $(a_{\eta })_{\eta \in 2^{<\omega }}$ so that:

• For all $\eta \in 2^{\omega }$ , $\{\varphi (x;a_{\eta  \alpha }) : \alpha < \omega \}$ is consistent.

• For all $\eta \in 2^{<\omega }$ , if $\nu $ extends $\eta \frown \langle 0 \rangle $ , then $\{\varphi (x;a_{\nu }), \varphi (x;a_{\eta \frown 1})\}$ is inconsistent,
where $\unlhd $ denotes the tree partial order on $2^{<\omega }$ . We say T is SOP $_{1}$ if some formula has SOP $_{1}$ modulo T. T is NSOP $_{1}$ otherwise.
Definition 2.6. Suppose A is a set of parameters.

(1) We say that tuples a and b have the same (Shelah) strong type over A, written $a \equiv ^{S}_{A} b$ , if $E(a,b)$ holds (i.e., $E(a',b')$ holds for all corresponding finite subtuples $a'$ and $b'$ of a and b, respectively) for every Adefinable equivalence relation $E(x,y)$ with finitely many classes.

(2) The group $\mathrm {Autf}(\mathbb {M}/A)$ of Lascar strong automorphisms (of the monster) over A is the subgroup of $\mathrm {Aut}(\mathbb {M}/A)$ generated by $\bigcup \{\mathrm {Aut}(\mathbb {M}/M) : A \subseteq M \prec \mathbb {M} \}$ . We say a and b have the same Lascar strong type over A, written $a \equiv ^{L}_{A} b$ , if there is some $\sigma \in \mathrm {Autf}(\mathbb {M}/A)$ such that $\sigma (a) = b$ . By a Lascar strong type over A, we mean an equivalence class of the relation $\equiv ^{L}_{A}$ .

(3) A typedefinable equivalence relation E on $\alpha $ tuples, for an ordinal $\alpha $ , is called bounded if it has small number of classes. We say a and b have the same KPstrong type over A, written $a \equiv ^{\mathrm {KP}}_{A} b$ , if $E(a,b)$ holds for all bounded typedefinable equivalence relations over A.

(4) We say that T is Gcompact over A when $a \equiv ^{L}_{A} b$ if and only if $a \equiv ^{\mathrm {KP}}_{A} b$ for all (possibly infinite) tuples $a,b$ . We say T is Gcompact if it is Gcompact over all finite sets A.
In [Reference Dobrowolski, Kim and Ramsey5], several basic facts about Kimindependence in NSOP $_{1}$ theories with existence were established. As we will make extensive use of them throughout the paper, we record them below.
Fact 2.7. Assume T is NSOP $_{1}$ with existence and A is a set of parameters. Then the following properties hold.

(1) Extension: If $\pi (x)$ is a partial type over $B \supseteq A$ which does not Kimdivide over A, then there is a completion $p \in S(B)$ of $\pi $ that does not Kimdivide over A. In particular, if and c is arbitrary, there is some $a' \equiv _{Ab} a$ such that [Reference Dobrowolski, Kim and Ramsey5, Proposition 4.1].

(2) Symmetry: [Reference Dobrowolski, Kim and Ramsey5, Corollary 4.9].

(3) Kim’s lemma for Morley sequences: The formula $\varphi (x;a)$ Kimdivides over A if and only if $\{\varphi (x;a_{i}) : i < \omega \}$ is inconsistent for all Morley sequences $\langle a_{i} : i < \omega \rangle $ over A with $a_{0} = a$ [Reference Dobrowolski, Kim and Ramsey5, Theorem 3.5].

(4) Kimforking = Kimdividing: If a formula $\varphi (x;a)$ Kimforks over A, then $\varphi (x;a)$ Kimdivides over A [Reference Dobrowolski, Kim and Ramsey5, Proposition 4.1].

(5) The chain condition: If and $I = \langle b_{i} : i < \omega \rangle $ is a Morley sequence over A with $b_{0} = b$ , then there is $a' \equiv _{Ab} a$ such that I is $Aa'$ indiscernible and (this follows from (3), as in, e.g., [Reference Dobrowolski, Kim and Ramsey5, Corollary 5.15]).

(6) The independence theorem for Lascar strong types: If $a_{0} \equiv ^{L}_{A} a_{1}$ , , , and , then there is some $a_{*}$ with $a_{*} \equiv ^{L}_{Ab} a_{0}$ , $a_{*} \equiv ^{L}_{Ac} a_{1}$ , and [Reference Dobrowolski, Kim and Ramsey5, Theorem 5.8].

(7) $T_{A}$ is Gcompact for any small set A, where $T_{A}$ is the theory of the monster model in the language with constants for the elements of A [Reference Dobrowolski, Kim and Ramsey5, Corollary 5.9].
As these facts make up part of the standard tool box for reasoning about Kimindependence, we will often make implicit use of these properties. For example, Kim’s lemma for Morley sequences, item (3) in the above list, is often used in this paper in the following way: if $I = \langle a_{i} : i < \omega \rangle $ is a Morley sequence over A with $a_{0} = a$ which is $Ab$ indiscernible, then . To see this, by symmetry (Item (2)), it suffices to show that which, by item (4), means that we need to show that there is no formula $\varphi (x;a) \in \text {tp}(b/Aa)$ which Kimdivides over A. But if $\varphi (x;a)$ Kimdivides over A, then Kim’s lemma implies that $\{\varphi (x;a_{i}) : i < \omega \}$ is inconsistent. This set of formulas, however, is realized by b so there can be no such formula.
The following is local character of Kimindependence for NSOP $_{1}$ theories. The usual formulation of local character for nonforking independence in simple theories merely asserts that, for any type $p \in S(A)$ , the set of B with $B \leq T$ such that p does not fork over B is nonempty, but it follows by base monotonicity, then, that p does not fork over C for any $B \subseteq C \subseteq A$ . Because Kimindependence, in general, does not satisfy base monotonicity in NSOP $_{1}$ theories, the following is the appropriate analogue for this setting:
Fact 2.8 [Reference Kaplan, Ramsey and Shelah9, Theorem 3.9].
Suppose T is NSOP $_{1}$ and $M \models T$ with $M \geq T$ . Given any $p \in S(M)$ (in finitely many variables), the set X defined by
satisfies the following:

(1) X is closed: if $\langle N_{i} : i < T \rangle $ is a sequence of models in X with $N_{i} \subseteq N_{j}$ for all $i < j$ , then $\bigcup _{i < T} N_{i} \in X$ .

(2) X is unbounded: if $Y \subset M$ has cardinality $\leq T$ , there is some $N \in X$ with $Y \subseteq N$ .
Remark 2.9. It is an easy consequence of Fact 2.8 that if $M \models T$ is equal to the union of $\langle N_{i} : i < T^{+}\rangle $ , an increasing and continuous (i.e., $N_{\delta } = \bigcup _{i < \delta } N_{i}$ for all limit $\delta $ ) elementary chain of models of T of size $T$ , then for any $p \in S(M)$ , there is some $i < T^{+}$ such that p does not Kimdivide over $N_{i}$ .
2.1 Trees
At several points in the paper, we will construct indiscernible sequences by an inductive construction of indiscernible trees. We recall the basic framework for these ‘treeinductions’ from [Reference Kaplan and Ramsey7]. For an ordinal $\alpha $ , let the language $L_{s,\alpha }$ be $\langle \unlhd , \wedge , <_{lex}, (P_{\beta })_{\beta \leq \alpha } \rangle $ . For us, a tree will mean a partial order $\unlhd $ such that for all x, the elements $\{y : y \unlhd x\}$ below x are linearly ordered (and not necessarily wellordered) by $\unlhd $ and such that for all $x,y$ , x and y have an infimum, i.e., there is a $\unlhd $ greatest element $z \unlhd x,y$ , which is called the meet of x and y. We may view a tree with $\alpha $ levels as an $L_{s,\alpha }$ structure by interpreting $\unlhd $ as the tree partial order, $\wedge $ as the binary meet function, $<_{lex}$ as the lexicographic order, and $P_{\beta }$ interpreted to define level $\beta $ . The specific trees, and the interpretations of these symbols that turn them into $L_{s,\alpha }$ structures, that we will need in this paper are outlined precisely in Definition 2.12.
We now recall the modeling property. In what follows, we will write $\mathrm {qftp}_{L'}(a)$ to denote the quantifierfree type of a in the language $L'$ and write $\text {tp}_{\Delta }(b/A)$ to denote the $\Delta $ type of b over A (i.e., the set of positive and negative instances of formulas in $\Delta $ with parameters from A satisfied by b). Although the subscript is used in two conflicting ways, it will be clear from context which is intended.
Definition 2.10. Suppose I is an $L'$ structure, where $L'$ is some language.

(1) We say $(a_{i} : i \in I)$ is a set of Iindexed indiscernibles over A if whenever
$(s_{0}, \ldots , s_{n1})$ , $(t_{0}, \ldots , t_{n1})$ are tuples from I with
$$ \begin{align*} \text{qftp}_{L'}(s_{0}, \ldots, s_{n1}) = \text{qftp}_{L'}(t_{0}, \ldots, t_{n1}), \end{align*} $$then we have$$ \begin{align*} \text{tp}(a_{s_{0}},\ldots, a_{s_{n1}}/A) = \text{tp}(a_{t_{0}},\ldots, a_{t_{n1}}/A). \end{align*} $$ 
(2) In the case that $L' = L_{s,\alpha }$ for some $\alpha $ , we say that an Iindexed indiscernible is sindiscernible. As the only $L_{s,\alpha }$ structures we will consider will be trees, we will often refer to Iindexed indiscernibles in this case as sindiscernible trees.

(3) We say that Iindexed indiscernibles have the modeling property if, given any $(a_{i} : i \in I)$ from $\mathbb {M}$ and any A, there is an Iindexed indiscernible $(b_{i} : i \in I)$ over A in $\mathbb {M}$ locally based on $(a_{i} : i \in I)$ over A. That is, given any finite set of formulas $\Delta $ from $L(A)$ and a finite tuple $(t_{0}, \ldots , t_{n1})$ from I, there is a tuple $(s_{0}, \ldots , s_{n1})$ from I so that
$$\begin{align*}\text{qftp}_{L'} (t_{0}, \ldots, t_{n1}) =\text{qftp}_{L'}(s_{0}, \ldots , s_{n1}) \end{align*}$$and also$$\begin{align*}\text{tp}_{\Delta}(b_{t_{0}}, \ldots, b_{t_{n1}}) = \text{tp}_{\Delta}(a_{s_{0}}, \ldots, a_{s_{n1}}). \end{align*}$$
Fact 2.11 [Reference Kim, Kim and Scow14, Theorem 4.3].
Let $I_{s}$ denote the $L_{s,\omega }$ structure
with all symbols being given their intended interpretations and each $P_{\alpha }$ naming the elements of the tree at level $\alpha $ . Then $I_{s}$ indexed indiscernibles have the modeling property.
Our trees will be understood to be an $L_{s,\alpha }$ structure for some appropriate $\alpha $ . As in [Reference Kaplan and Ramsey7], we introduce a distinguished class of trees $\mathcal {T}_{\alpha }$ .
Definition 2.12. Suppose $\alpha $ is an ordinal. We define $\mathcal {T}_{\alpha }$ to be the set of functions f such that:

• $\text {dom}(f)$ is an endsegment of $\alpha $ of the form $[\beta ,\alpha )$ for $\beta $ equal to $0$ or a successor ordinal. If $\alpha $ is a successor, we allow $\beta = \alpha $ , i.e., $\text {dom}(f) = \emptyset $ .

• $\text {ran}(f) \subseteq \omega $ .

• Finite support: The set $\{\gamma \in \text {dom}(f) : f(\gamma ) \neq 0\}$ is finite.
We interpret $\mathcal {T}_{\alpha }$ as an $L_{s,\alpha }$ structure by defining:

• $f \unlhd g$ if and only if $f \subseteq g$ . Write $f \perp g$ if $\neg (f \unlhd g)$ and $\neg (g \unlhd f)$ .

• $f \wedge g = f_{[\beta , \alpha )} = g_{[\beta , \alpha )}$ where $\beta = \text {min}\{ \gamma : f_{[\gamma , \alpha )} =g_{[\gamma , \alpha )}\}$ , if nonempty (note that $\beta $ will not be a limit, by finite support). Define $f \wedge g$ to be the empty function if this set is empty (note that this cannot occur if $\alpha $ is a limit).

• $f <_{lex} g$ if and only if $f \vartriangleleft g$ or, $f \perp g$ with $\text {dom}(f \wedge g) = [\gamma +1,\alpha )$ and $f(\gamma ) < g(\gamma ).$

• For all $\beta \leq \alpha $ , $P_{\beta } = \{ f \in \mathcal {T}_{\alpha } : \text {dom}(f) = [\beta , \alpha )\}$ . Note that $P_{0}$ are the leaves of the tree (i.e., the top level) and $P_{\alpha }$ is empty for $\alpha $ limit.
Fact 2.11 and compactness can be used to show that $\mathcal {T}_{\alpha }$ indexed indiscernibles have the modeling property as well [Reference Kaplan and Ramsey7, Corollary 5.6].
Definition 2.13. Suppose $\alpha $ is an ordinal.

(1) (Restriction) If $v \subseteq \alpha $ , the restriction of $\mathcal {T}_{\alpha }$ to the set of levels v is the $L_{s,\alpha }$ substructure of $\mathcal {T}_{\alpha }$ with the following underlying set:
$$ \begin{align*} \mathcal{T}_{\alpha} \upharpoonright v = \{\eta \in \mathcal{T}_{\alpha} : \min (\text{dom}(\eta)) \in v \text{ and }\beta \in \text{dom}(\eta) \setminus v \implies \eta(\beta) = 0\}. \end{align*} $$ 
(2) (Concatenation) If $\eta \in \mathcal {T}_{\alpha }$ , $\text {dom}(\eta ) = [\beta +1,\alpha )$ for $\beta $ nonlimit, and $i < \omega $ , let $\eta \frown \langle i \rangle $ denote the function $\eta \cup \{(\beta ,i)\}$ . We define $\langle i \rangle \frown \eta \in \mathcal {T}_{\alpha +1}$ to be $\eta \cup \{(\alpha ,i)\}$ . We write $\langle i \rangle $ for $\emptyset \frown \langle i \rangle $ .

(3) (Canonical inclusions) If $\alpha < \beta $ , we define the map $\iota _{\alpha \beta } : \mathcal {T}_{\alpha } \to \mathcal {T}_{\beta }$ by $\iota _{\alpha \beta }(f) := f \cup \{(\gamma , 0) : \gamma \in \beta \setminus \alpha \}$ .

(4) (The all $0$ ’s path) If $\beta < \alpha $ , then $\zeta _{\beta }$ denotes the function with $\text {dom}(\zeta _{\beta }) = [\beta , \alpha )$ and $\zeta _{\beta }(\gamma ) = 0$ for all $\gamma \in [\beta ,\alpha )$ . This defines an element of $\mathcal {T}_{\alpha }$ if and only if $\beta \in \{\gamma \in \alpha \mid \gamma \mbox { is not limit}\}=:[\alpha ]$ .

(5) (Tuple notation) Given $\nu \in \mathcal {T}_{\alpha }$ , we write $a_{\unrhd \nu }$ for the tuple enumerating $\{a_{\xi } : \nu \unlhd \xi \in \mathcal {T}_{\alpha }\}$ .
In previous works on Kimindependence over arbitrary sets, there was a gap concerning the construction of Morley trees (and a parallel gap in the theory over models), first discovered by Jan Dobrowolski and Mark Kamsma. Namely, there is no reason a priori for an sindiscernible tree locally based on a weakly spread out tree (see Definition 2.17) to be weakly spread out, which is needed to continue the induction. Over models this has a very easy fix: one can check that it is possible to choose a global Minvariant type $q \supseteq \text {tp}((a_{\eta })_{\eta \in \mathcal {T}_{\alpha }}/M)$ such that, if $(a^{\prime }_{\eta })_{\eta \in \mathcal {T}_{\alpha }} \models q$ then $(a^{\prime }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ is $\mathbb {M}$ indiscernible. Morley sequences in such types allow the argument to work without change (the details for this case will appear elsewhere). But over sets, a lengthier argument is required to patch the proofs. The relevant notion for the modification is that of a mutually sindiscernible sequence. We prove in Lemma 2.15 that, given an sindiscernible tree, there is a Morley sequence starting with this tree which is mutually sindiscernible, and then we show in Lemma 2.16 that this notion is preserved upon passage to an sindiscernible tree.
Definition 2.14. We say a sequence $\langle (a_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }} : i < \kappa \rangle $ is mutually sindiscernible over A if, for all $i < \kappa $ , $(a_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }}$ is sindiscernible over $A\{a_{\eta ,j} : \eta \in \mathcal {T}_{\alpha }, j \neq i \}$ .
Lemma 2.15. Assume A is an extension base. Given a tree $(a_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ that is sindiscernible over A, there is a sequence $I = \langle (a_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }} : i < \omega \rangle $ such that $(a_{\eta ,0})_{\eta \in \mathcal {T}_{\alpha }} = (a_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ , I is a Morley sequence over A, and I is mutually sindiscernible over A.
Proof Let $\kappa $ be sufficiently large with respect to $A$ . By induction on $\gamma < \kappa $ , we will choose $(a_{\eta ,\gamma })_{\eta \in \mathcal {T}_{\alpha }}$ such that, taking $I_{\gamma }$ to be the sequence $\langle (a_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }} : i < \gamma \rangle $ , we have that $I_{\gamma }$ starts with $(a_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ , is mutually sindiscernible over A, and satisfies
for all $i < \gamma $ . The sequence $I_{1}$ is already specified and trivially satisfies the requirements.
Assume we are given $(a_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }}$ for all $i < \gamma $ and set $I_{\gamma } = \langle (a_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }} : i < \gamma \rangle $ . Apply extension to get some $(b_{\eta })_{\eta \in \mathcal {T}_{\alpha }} \equiv _{A} (a_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ such that
By the modeling property, we can take $(b^{\prime }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ to be locally based on $(b_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ and sindiscernible over $AI_{\gamma }$ , then we we still have $(b^{\prime }_{\eta })_{\eta \in \mathcal {T}_{\alpha }} \equiv _{A} (a_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ , as $(a_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ was assumed to be sindiscernible over A, and local basedness and strong finite character of nonforking imply
Now by induction on $i < \gamma $ , we will choose $(a^{\prime }_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }}$ satisfying the following conditions:

(1) $(a^{\prime }_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }}$ is sindiscernible over
$$ \begin{align*}A \cup \{a^{\prime}_{\eta,j} : \eta \in \mathcal{T}_{\alpha}, j < i\} \cup \{a_{\eta,k} : \eta \in \mathcal{T}_{\alpha}, k> i\} \cup \{b^{\prime}_{\eta} : \eta \in \mathcal{T}_{\alpha}\}. \end{align*} $$ 
(2) $(b^{\prime }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ is sindiscernible over
$$ \begin{align*}A \cup \{a^{\prime}_{\eta,j} : \eta \in \mathcal{T}_{\alpha}, j \leq i\} \cup \{a_{\eta,k} : \eta \in \mathcal{T}_{\alpha}, k> i\}. \end{align*} $$ 
(3) $(a^{\prime }_{\eta ,j})_{\eta \in \mathcal {T}_{\alpha }, j \leq i} (a_{\eta ,k})_{\eta \in \mathcal {T}_{\alpha }, k> i} \equiv _{A} (a_{\eta ,j})_{\eta \in \mathcal {T}_{\alpha }, j \leq i} (a_{\eta ,k})_{\eta \in \mathcal {T}_{\alpha }, k > i}$ .

(4) The following independence holds:
Fix $i < \gamma $ and suppose we have chosen $(a^{\prime }_{\eta ,j})_{\eta \in \mathcal {T}_{\alpha }}$ for all $j < i$ . Pick $(a^{\prime }_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }}$ sindiscernible over $A \cup \{a^{\prime }_{\eta ,j} : \eta \in \mathcal {T}_{\alpha }, j < i\} \cup \{a_{\eta ,k} : \eta \in \mathcal {T}_{\alpha }, k> i\} \cup \{b^{\prime }_{\eta } : \eta \in \mathcal {T}_{\alpha }\}$ and locally based on $(a_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }}$ . Then (1) is satisfied and (2) is easy to check using local basedness and the inductive assumption. We assumed $I_{\gamma }$ was mutually sindiscernible over A and hence by (3) of the inductive hypothesis, we know that $(a_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }}$ is sindiscernible over
and therefore $(a^{\prime }_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }}$ has the same type over this set, which establishes (3). Finally, (4) follows by local basedness, (3), and the invariance of nonforking independence. More explicitly, suppose there is a finite tuple b from $(b^{\prime }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ , a finite tuple a from $\{a^{\prime }_{\eta ,j} : j < i\} \cup \{a_{\eta ,k} : k> i\}$ , and a finite tuple $\overline {\eta }$ from $\mathcal {T}_{\alpha }$ such that
where $\varphi (x;y,z) \in L(A)$ is a formula such that $\varphi (x;a^{\prime }_{\overline {\eta },i},a)$ forks over A. Local basedness entails that there is $\overline {\nu }$ with $\mathrm {qftp}_{L_{s,\alpha }}(\overline {\nu }) = \mathrm {qftp}_{L_{s,\alpha }}(\overline {\eta })$ such that
But by mutual sindiscernibility, $a_{\overline {\nu },i} \equiv _{Aa} a_{\overline {\eta },i}$ and, by (3), $a_{\overline {\eta },i} \equiv _{Aa} a^{\prime }_{\overline {\eta },i}$ and hence $\varphi (x;a_{\overline {\nu },i},a)$ forks over A as well. This contradicts the inductive hypothesis that
This shows that our choice of $(a^{\prime }_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }}$ satisfies the requirements.
Having constructed our sequence $I^{\prime }_{\gamma } = \langle (a^{\prime }_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }} : i < \gamma \rangle $ , we have $I^{\prime }_{\gamma } \equiv _{A} I_{\gamma }$ by (3) and $(b^{\prime }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ is sindiscernible over $I^{\prime }_{\gamma }$ by (2). Moreover, each $(a_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }}$ is indiscernible over $A(b^{\prime }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}(a_{\eta ,j})_{\eta \in \mathcal {T}_{\alpha },j \neq i}$ by (1). Finally, by (4), we have . Choosing $(a_{\eta ,\gamma })$ such that
we arrive at $I_{\gamma +1}$ . There is nothing to do at limits, so we have succeeded in constructing our sequence $I_{\kappa }$ . Applying Erdős–Rado to $I_{\kappa }$ , then, we obtain the desired sequence I.
Lemma 2.16. Suppose $(a_{\eta })_{\eta \in \mathcal {T}_{\alpha +1}}$ is a tree of tuples such that $I = \langle a_{\unrhd \langle i \rangle } : i < \omega \rangle $ is mutually sindiscernible and Morley over A. Then if $(a^{\prime }_{\eta })_{\eta \in \mathcal {T}_{\alpha +1}}$ is sindiscernible and locally based on $(a_{\eta })_{\eta \in \mathcal {T}_{\alpha +1}}$ over A and $I' = \langle a^{\prime }_{\unrhd \langle i \rangle }: i < \omega \rangle $ , then $I' \equiv _{A} I$ and thus $I'$ is also mutually sindiscernible and Morley over A.
Proof Suppose $\overline {\eta }$ and $\overline {\nu }$ are tuples from $\mathcal {T}_{\alpha +1} \setminus \{\emptyset \}$ with $\mathrm {qftp}_{L_{s,\alpha +1}}(\overline {\eta }) = \mathrm {qftp}_{L_{s,\alpha +1}}(\overline {\nu })$ . After possibly reordering the tuples, there are $i_{0} < \cdots < i_{k1}$ and $j_{0} < \cdots < j_{k1}$ such that $\overline {\eta } = (\overline {\eta }_{0}, \ldots , \overline {\eta }_{k1})$ and $\overline {\nu } = (\overline {\nu }_{0}, \ldots , \overline {\nu }_{k1})$ where each $\overline {\eta }_{l}$ comes from the tree $\unrhd \langle i_{l} \rangle $ and $\overline {\nu }_{l}$ comes from the tree $\unrhd \langle j_{l} \rangle $ for $l < k$ . Then, in particular, $\mathrm {qftp}_{L_{s,\alpha +1}}(\overline {\eta }_{l}) = \mathrm {qftp}_{L_{s,\alpha +1}}(\overline {\nu }_{l})$ for all $l < k$ . Additionally, for all $l < k$ , let $\overline {\eta }^{\prime }_{l}$ be the element of the tree $\unrhd \langle j_{l} \rangle $ corresponding to $\overline {\eta }_{l}$ (i.e., replace each node $\langle i_{l}\rangle ^{\frown } \xi $ enumerated in $\overline {\eta }_{l}$ with $\langle j_{l} \rangle ^{\frown } \xi $ ). Because I is an Aindiscernible sequence, we have
Additionally, in the tree $\unrhd \langle j_{l} \rangle $ (naturally viewed as an $L_{s,\alpha }$ structure), we have $\mathrm {qftp}_{L_{s,\alpha }}(\overline {\eta }_{l}') = \mathrm {qftp}_{L_{s,\alpha }}(\overline {\eta }_{l})$ for all $l < k$ . Thus, mutual sindiscernibility entails
Thus we have shown that $a_{\overline {\eta }} \equiv _{A} a_{\overline {\nu }}$ . Therefore, it follows, by local basedness, that $I \equiv _{A} I'$ and the result follows.
Definition 2.17. Suppose $(a_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ is a tree of tuples in $\mathbb {M}$ , and A is a set of parameters.

(1) We say $(a_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ is weakly spread out over A if for all $\eta \in \mathcal {T}_{\alpha }$ with $\text {dom}(\eta ) =[\beta +1,\alpha )$ for some $\beta \in [\alpha ]$ , the sequence of cones $(a_{\unrhd \eta \frown \langle i \rangle })_{i < \omega }$ is a Morley sequence in $\mathrm {tp}(a_{\unrhd \eta \frown \langle 0 \rangle }/A)$ .

(2) Suppose $(a_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ is a tree which is weakly spread out and sindiscernible over A and for all pairs of finite subsets $w,v$ of $\alpha $ with $w = v$ ,
$$ \begin{align*}(a_{\eta})_{\eta \in \mathcal{T}_{\alpha} \upharpoonright w} \equiv_{A} (a_{\eta})_{\eta \in \mathcal{T}_{\alpha} \upharpoonright v} \end{align*} $$then we say $(a_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ is a weakly Morley tree over A. 
(3) A weak tree Morley sequence over A is a Aindiscernible sequence of the form $(a_{\zeta _\beta })_{\beta \in [\alpha ]}$ for some weakly Morley tree $(a_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ over A. More generally, we will say an Aindiscernible sequence I is a weak tree Morley sequence over A if it is EMequivalent to a sequence of this form.
Remark 2.18. If $I = \langle b_{i} : i < \omega \rangle $ is an Aindiscernible sequence and $I \equiv _{A} J$ for some weak tree Morley sequence J over A, then I is a weak tree Morley sequence over A. In particular, if I is a subsequence of J, by the Aindiscernibility of J, the sequence I is also weak tree Morley over A.
Fact 2.19. Suppose T is NSOP $_{1}$ with existence and A is a set of parameters.

(1) If , there is an $Ab$ indiscernible sequence $I = \langle a_{i} : i < \omega \rangle $ over A with $a_{0} = a$ such that I is weak tree Morley over A [Reference Dobrowolski, Kim and Ramsey5, Lemma 4.7].

(2) Kim’s lemma for weak tree Morley sequences: the formula $\varphi (x;a_{0})$ Kimdivides over A if and only if $\{\varphi (x;a_{i}) : i < \omega \}$ is inconsistent for some weak tree Morley sequence $\langle a_{i} : i < \omega \rangle $ over A if and only if $\{\varphi (x;a_{i}) : i < \omega \}$ is inconsistent for all weak tree Morley sequences $\langle a_{i} : i < \omega \rangle $ over A [Reference Dobrowolski, Kim and Ramsey5, Corollary 4.8].

(3) If $a \equiv ^{L}_{A} b$ and , there is an Morley sequence over A starting with $(a,b)$ as its first two elements (follows from Fact 2.7 as in [Reference Kaplan and Ramsey7, Corollary 6.6]).
2.2 Further properties of Kimindependence
Fact 2.20 [Reference Dobrowolski, Kim and Ramsey5, Lemma 5.7].
Suppose T is NSOP $_{1}$ with existence. If A is a set of parameters, c is an arbitrary tuple, and , then there is $a' \equiv ^{L}_{Ab} a$ such that .
The following lemma is easy and wellknown, but, in the absence of a clear reference, we provide a proof:
Lemma 2.21. Suppose T is NSOP $_{1}$ with existence, A is a set of parameters, and $p(x) \in S(A)$ .

(1) Given any tuple of variables y, there is a partial type $\Gamma (x,y)$ over A such that $(a,b) \models \Gamma (x,y)$ if and only if $a \models p$ and .

(2) There is a partial type $\Delta (x_{i} : i < \omega )$ over A such that $I = \langle a_{i} : i < \omega \rangle \models \Delta $ if and only if I is an Morley sequence over A in p.
Proof (1) By compactness, we may assume y is finite. Fix $c \models p$ and define $\Gamma (x,y)$ by
By symmetry, invariance, and Kimforking = Kimdividing, this partial type is as desired.
(2) One can take $\Delta $ to be the partial type that asserts $\langle x_{i} : i < \omega \rangle $ is Aindiscernible, every $x_{i} \models p$ , and (which is typedefinable over A by (1)).
The following lemma is the analogue of the ‘strong independence theorem’ of [Reference Kruckman and Ramsey16, Theorem 2.3] for Lascar strong types.
Lemma 2.22. Suppose T is NSOP $_{1}$ with existence. If A is a set of parameters, , , , and $a_{0} \equiv ^{L}_{A} a_{1}$ , then there is a such that $a \equiv ^{L}_{Ab} a_{0}$ , $a \equiv ^{L}_{Ac} a_{1}$ and, additionally, we have , , and .
Proof By Fact 2.20, there is $c' \equiv ^{L}_{Ab} c$ such that . Let $\sigma \in \mathrm {Autf}(\mathbb {M}/Ab)$ be an automorphism such that $\sigma (c') = c$ and let $c_{0} = \sigma (c)$ . Then we have and $c_{0} \equiv _{Ab}^{L} c$ and hence, in particular, $c_{0}b \equiv _{A}^{L} cb$ . By symmetry and a second application of Fact 2.20 once again, we find $b"c" \equiv ^{L}_{Ac} bc_{0}$ with . Let $\tau \in \mathrm {Autf}(\mathbb {M}/Ac)$ be a strong automorphism with $\tau (b"c") = bc_{0}$ and define $b_{1} = \tau (b)$ . Then by construction, we have $b"c" \equiv _{A} bc_{0}$ and $bc_{0} \equiv ^{L}_{A} bc$ , it follows that $b"c" \equiv ^{L}_{A} bc$ , and hence $\tau (b"c") \equiv ^{L}_{A} \tau (bc)$ , which, after unraveling definitions, gives $bc_{0} \equiv ^{L}_{A} b_{1}c$ . Moreover, since , we obtain by invariance. Let $b_{0} = b$ and $c_{1} = c$ . By Fact 2.19(3), we can extend the sequence $\langle (b_{i},c_{i}) : i < 2 \rangle $ to a weak tree Morley sequence $I = \langle (b_{i},c_{i}) : i \in \mathbb {Z} \rangle $ over A.
Choose $a'$ such that $a_{1}c_{1} \equiv ^{L}_{A} a'c_{0}$ . Then we have $a_{0} \equiv ^{L}_{A} a'$ , as well as , by our assumptions. Additionally, since , $b_{0} = b$ and $c \equiv _{Ab} c_{0}$ , we have . Therefore, by Fact 2.7(6), there is $a_{*}$ with $a_{*} \equiv ^{L}_{Ab_{0}} a_{0}$ , $a_{*} \equiv ^{L}_{Ac_{0}} a'$ , with .
Because I is a weak tree Morley sequence and , by Kim’s lemma, compactness, and an automorphism, there is such that $a_{**}b_{0}c_{0} \equiv ^{L}_{A} a_{*}b_{0}c_{0}$ and such that I is $Aa_{**}$ indiscernible. Note that, by construction, $a_{**} \equiv ^{L}_{Ab} a_{0}$ , $a_{**} \equiv ^{L}_{Ac} a_{1}$ , and .
Additionally, the sequence $\langle b_{i} : i \in \mathbb {Z}^{\leq 0} \rangle $ is a weak tree Morley sequence over A which is $Aa_{**}c$ indiscernible and containing $b_{0} = b$ , hence , by Kim’s lemma for weak tree Morley sequences. Similarly, the sequence $\langle c_{i} : i \in \mathbb {Z}^{\geq 1} \rangle $ is a weak tree Morley sequence over A containing $c = c_{1}$ which is $Aa_{**}b$ indiscernible, yielding . By symmetry, we conclude.
3 Transitivity and witnessing
3.1 Preliminary lemmas
We begin by establishing some lemmas, allowing us to construct sequences that are Morley over more than one base simultaneously. The broad structure of the argument will follow that of [Reference Kaplan and Ramsey8], which established transitivity over models for Kimindependence in NSOP $_{1}$ theories, however, all uses of coheirs and heirs will need to be replaced.
In particular, the following lemma does not follow the corresponding [Reference Kaplan and Ramsey8, Lemma 3.1], instead producing the desired sequence by a treeinduction.
Lemma 3.1. Suppose T is NSOP $_{1}$ and satisfies the existence axiom. If $A \subseteq B$ and , then there is a weak tree Morley sequence $\langle a_{i} : i < \omega \rangle $ over B with $a_{0} = a$ such that for all $i < \omega $ .
Proof By induction on $\alpha $ , we will construct trees $(a^{\alpha }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ so that:

(1) $(a^{\alpha }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ is sindiscernible and weakly spread out over B.

(2) $a^{\alpha }_{\eta } \models \text {tp}(a/B)$ for all $\eta \in \mathcal {T}_{\alpha }$ .

(3) If $\alpha $ is a successor, .

(4) If $\alpha < \beta $ , then $a^{\beta }_{\iota _{\alpha \beta }(\eta )} = a^{\alpha }_{\eta }$ for all $\eta \in \mathcal {T}_{\alpha }$ .
For $\alpha = 0$ , we put $a^{0}_{\emptyset } = a$ , and for $\delta $ limit, we will define $(a^{\delta }_{\eta })_{\eta \in \mathcal {T}_{\delta }}$ by setting $a^{\delta }_{\iota _{\alpha \delta }(\eta )} = a^{\alpha }_{\eta }$ for all $\alpha < \delta $ and $\eta \in \mathcal {T}_{\alpha }$ which, by (4) and induction, is welldefined and satisfies the requirements.
Now suppose we are given $(a^{\beta }_{\eta })_{\eta \in \mathcal {T}_{\beta }}$ satisfying the requirements for all $\beta \leq \alpha $ . Let $\langle (a^{\alpha }_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }} : i < \omega \rangle $ be a mutually sindiscernible Morley sequence over B with $a^{\alpha }_{\eta ,0} = a^{\alpha }_{\eta }$ for all $\eta \in \mathcal {T}_{\alpha }$ , which exists by Lemma 2.15. Apply extension to find $a_{*} \equiv _{B} a$ so that . Define a tree $(b_{\eta })_{\eta \in \mathcal {T}_{\alpha +1}}$ by setting $b_{\emptyset }= a_{*}$ and $b_{\langle i \rangle \frown \eta } = a^{\alpha }_{\eta ,i}$ for all $i < \omega $ and $\eta \in \mathcal {T}_{\alpha }$ . We may define $(a^{\alpha +1}_{\eta })_{\eta \in \mathcal {T}_{\alpha +1}}$ to be a tree which is sindiscernible over B and locally based on $(b_{\eta })_{\eta \in \mathcal {T}_{\alpha +1}}$ over B. By an automorphism, we may assume $a^{\alpha +1}_{\iota _{\alpha \alpha +1}(\eta )} = a^{\alpha }_{\eta }$ for all $\eta \in \mathcal {T}_{\alpha }$ , hence conditions (2), and (4) are clearly satisfied. Moreover, by Lemma 2.16, we have $\langle (a_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }} : i < \omega \rangle \equiv _{B} \langle a^{\alpha +1}_{\unrhd \langle i \rangle } : i < \omega \rangle $ so $\langle a^{\alpha +1}_{\unrhd \langle i \rangle } : i < \omega \rangle $ is a Morley sequence over B. Then by (4) and induction, it follows that $(a^{\alpha +1}_{\eta })_{\eta \in \mathcal {T}_{\alpha +1}}$ is sindiscernible and spread out over B, which shows (1).
For (3), we just note that, by symmetry, if , there is some formula $\varphi (x;a_{\emptyset }^{\alpha +1}) \in \text {tp}(Ba^{\alpha +1}_{\vartriangleright \emptyset }/Aa^{\alpha +1}_{\emptyset })$ that Kimdivides over A. As the tree $(a^{\alpha +1}_{\eta })_{\eta \in \mathcal {T}_{\alpha +1}}$ is locally based on $(b_{\eta })_{\eta \in \mathcal {T}_{\alpha +1}}$ , it follows that some tuple from $Bb_{\vartriangleright \emptyset }$ also realizes $\varphi (x;a_{*})$ and $a_{*} \equiv _{B} a^{\alpha +1}_{\emptyset }$ , so $\varphi (x;a_{*})$ Kimdivides over A as well, contradicting the choice of $a_{*}$ . This contradiction establishes (3), completing the induction.
By considering $(a^{\kappa }_{\eta })_{\eta \in \mathcal {T}_{\kappa }}$ for $\kappa $ sufficiently large, we may apply Erdős–Rado, as in [Reference Kaplan and Ramsey7, Lemma 5.10], to find the desired sequence.
The proof of the next lemma follows [Reference Kaplan and Ramsey8, Lemma 3.2].
Lemma 3.2. Suppose T is an NSOP $_{1}$ theory satisfying the existence axiom. If and , then there is $c'$ so that $c' \equiv _{Ab} c$ , , and .
Proof Define a partial type $\Gamma (x;b,a)$ over $Aab$ as follows:
Claim 1. If $\langle a_{i} : i < \omega \rangle $ is an $Ab$ indiscernible sequence satisfying $a_{0} = a$ and for all $i < \omega $ , then $\bigcup _{i < \omega } \Gamma (x;b,a_{i})$ is consistent.
Proof of claim
By induction on $n < \omega $ , we will find $c_{n} \equiv ^{L}_{A} c$ such that and $c_{n} \models \bigcup _{i < n} \Gamma (x;b,a_{i})$ . For $n = 0$ , we can put $c_{0} = c$ , since by assumption. Assume we have found $c_{n}$ , and, by Fact 2.20, choose $c'$ such that $c' \equiv ^{L}_{A} c$ and . Then $c' \equiv ^{L}_{A} c \equiv ^{L}_{A} c_{n}$ and, since , we may apply Lemma 2.22 to find $c_{n+1} \equiv ^{L}_{A} c$ such that $c_{n+1} \models \text {tp}(c_{n}/Aba_{<n}) \cup \text {tp}(c'/Aa_{n})$ and such that and , hence, in particular, $c_{n+1} \models \bigcup _{i < n+1} \Gamma (x;b,a_{i})$ . The claim follows by compactness.
Next we define a partial type $\Delta (x;b,a)$ as follows:
Claim 2. The set of formulas $\Delta (x;b,a)$ is consistent.
Proof of claim
Suppose not. Then because Kimforking and Kimdividing are the same in NSOP $_{1}$ with existence, there is some formula $\psi (x;b,a) \in L(Aab)$ such that
and $\psi (x;b,a)$ Kimdivides over $Ab$ . As , we know by Lemma 3.1 that there is a sequence $\langle a_{i} : i < \omega \rangle $ with $a_{0} = a$ which is a weak tree Morley sequence over $Ab$ and satisfies for all $i < \omega $ . Then by Claim 1, $\bigcup _{i < \omega } \Gamma (x;b,a_{i})$ is consistent. However, we have
and $\{\psi (x;b,a_{i}) : i < \omega \}$ is inconsistent, because weak tree Morley sequences witness Kimdividing. This contradiction proves the claim.
To conclude, we may take $c'$ to be any realization of $\Delta (x;b,a)$ .
The next proposition is a strengthening of Fact 2.19(1).
Proposition 3.3. Suppose T is an NSOP $_{1}$ theory satisfying the existence axiom. If , then there is a sequence $I = \langle a_{i} : i < \omega \rangle $ with $a_{0} = a$ such that I is a weak tree Morley sequence over A and an Morley sequence over $Ab$ .
Proof By induction on $\alpha $ , we will construct trees $(a^{\alpha }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ satisfying the following:

(1) For all $\eta \in \mathcal {T}_{\alpha }$ , $a^{\alpha }_{\eta } \models \text {tp}(a/Ab)$ .

(2) The tree $(a^{\alpha }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ is sindiscernible over $Ab$ and weakly spread out over A.

(3) If $\alpha $ is a successor, then .

(4) .

(5) If $\alpha < \beta $ , then $a^{\beta }_{\iota _{\alpha \beta }(\eta )} = a^{\alpha }_{\eta }$ for all $\eta \in \mathcal {T}_{\alpha }$ .
Put $a^{0}_{\emptyset } = a$ and for $\delta $ limit, if we are given $(a^{\alpha }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ for every $\alpha < \delta $ , we can define $(a^{\delta }_{\eta })_{\eta \in \mathcal {T}_{\delta }}$ by setting $a^{\delta }_{\iota _{\alpha \delta }(\eta )} = a^{\alpha }_{\eta }$ for all $\alpha < \delta $ and $\eta \in \mathcal {T}_{\alpha }$ , which is welldefined by (5) and is easily seen to satisfy the requirements.
Suppose now we are given $(a^{\alpha }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ . Let $J = \langle (a^{\alpha }_{\eta ,i})_{\eta \in \mathcal {T}_{\alpha }} : i < \omega \rangle $ be a mutually sindiscernible Morley sequence over A with $(a^{\alpha }_{\eta ,0})_{\eta \in \mathcal {T}_{\alpha }} = (a^{\alpha }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ , which exists by Lemma 2.15. By (4), symmetry, and the chain condition, Fact 2.7(5) we may assume J is $Ab$ indiscernible and . By Lemma 3.2, there is $a_{*} \equiv _{Ab} a$ such that and . After defining a tree $(c_{\eta })_{\eta \in \mathcal {T}_{\alpha +1}}$ by $c_{\emptyset } = a_{*}$ and $c_{\langle i \rangle \frown \eta } = a^{\alpha }_{\eta ,i}$ for all $\eta \in \mathcal {T}_{\alpha }$ , these conditions on $a_{*}$ imply that $(c_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ satisfy (3) and (4), respectively. Let $(a^{\alpha +1}_{\eta })_{\eta \in \mathcal {T}_{\alpha +1}}$ be any tree sindiscernible over $Ab$ locally based on $(c_{\eta })_{\eta \in \mathcal {T}_{\alpha }}$ over $Ab$ . This still satisfies (2) by Lemma 2.16. Moreover, as $c_{\unrhd \langle i \rangle } \models \mathrm {tp}((a^{\alpha }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}/Ab)$ for all $i < \omega $ , it follows from local basedness that $a^{\alpha +1}_{\unrhd \langle i \rangle } \models \mathrm {tp}((a^{\alpha }_{\eta })_{\eta \in \mathcal {T}_{\alpha }}/Ab)$ for all $i < \omega $ as well. Hence, by an automorphism over $Ab$ , we may assume $a^{\alpha +1}_{0 \frown \eta } = a^{\alpha }_{\eta }$ for all $\eta \in \mathcal {T}_{\alpha }$ , which ensures the constructed tree satisfies (5), and (1)–(4) are easy to verify.
Given $(a^{\kappa }_{\eta })_{\eta \in \mathcal {T}_{\kappa }}$ for $\kappa $ sufficiently large, we may, by Erdős–Rado (see, e.g., [Reference Kaplan and Ramsey7, Lemma 5.10]), obtain a weak Morley tree $(b_{\eta })_{\eta \in \mathcal {T}_{\omega }}$ over A satisfying (1)–(4). Then the sequence $I = \langle a_{i} : i < \omega \rangle $ defined by $a_{i} = b_{\zeta _{i}}$ for all $i < \omega $ is a weak tree Morley sequence over A, as it is a path in a weak Morley tree over A, but by (3), we have for all i, so I is Morley over $Ab$ as well.
3.2 Transitivity and witnessing
The following theorem establishes the transitivity of Kimindependence in NSOP $_{1}$ theories with existence.
Theorem 3.4. Suppose T is NSOP $_{1}$ with existence. Then if $A \subseteq B$ , and , then .
Proof By Proposition 3.3 and the assumption that , there is a sequence $I = \langle a_{i} : i < \omega \rangle $ with $a_{0} = a$ such that I is an Morley sequence over B and a weak tree Morley sequence over A. As , by symmetry, and I is Morley over B, there is $I' \equiv _{Ba}I$ such that $I'$ is $Bc$ indiscernible. Because $I'$ is also a weak tree Morley sequence over A, it follows by Kim’s lemma that . By symmetry, we conclude.
Proposition 3.5. Assume T is NSOP $_1$ with existence. The following are equivalent.

(1) .

(2) There is a model $M\supseteq A$ such that (or ) and .

(3) There is a model $M\supseteq A$ such that (or ) and .
Proof (1) $\Rightarrow $ (2) Since , there is a Morley sequence $I=\langle a_i : i<\omega \rangle $ over A with $a_0=a$ such that I is $Ab$ indiscernible. By [Reference Dobrowolski, Kim and Ramsey5, Lemma 2.17], there is a model N containing A such that and I is a coheir sequence over N. By compactness and extension we can clearly assume the length of I is arbitrarily large, and . Hence by the pigeonhole principle, there is an infinite subsequence J of I such that all the tuples in J have the same type over $Nb$ . Thus, for $a'\in J$ , we have and . Hence $M=f(N)$ is a desired model, where f is an $Ab$ automorphism sending $a'$ to a.
(2) $\Rightarrow $ (3) Clear.
(3) $\Rightarrow $ (1) follows from transitivity and symmetry of .
Proposition 3.6. Suppose T satisfies the existence axiom. The following are equivalent for a cardinal $\kappa \geq T$ :

(1) T is NSOP $_{1}$ .

(2) There is no increasing continuous sequence $\langle A_{i} : i < \kappa ^{+} \rangle $ of parameter sets and finite tuple d such that $A_{i} \leq \kappa $ and for all $i < \kappa ^{+}$ .

(3) There is no set A of parameters of size $\kappa ^{+}$ and $p(x) \in S(A)$ with x a finite tuple of variables such that for some increasing and continuous sequence of sets $\langle A_{i} : i < \kappa ^{+} \rangle $ with union A, we have $A_{i} \leq \kappa $ and p Kimdivides over $A_{i}$ for all $i < \kappa ^{+}$ .
Proof (1) $\implies $ (2) It suffices to show that, given any increasing continuous sequence $\langle A_{i} : i < \kappa ^{+} \rangle $ of parameter sets and tuple d such that $A_{i} \leq \kappa $ and for all $i < \kappa ^{+}$ , there is a continuous increasing sequence of models $\langle M_{i} : i < \kappa ^{+} \rangle $ and a finite tuple $d'$ such that $M_{i} \leq \kappa $ and for all $i < \kappa ^{+}$ . This follows from Fact 2.8, since the existence of such a sequence of models implies T has SOP $_{1}$ . Moreover, after naming constants, we may assume $\kappa = T$ .
So suppose we are given $\langle A_{i} : i < T^{+} \rangle $ , an increasing continuous sequence of sets of parameters with $A_{i} \leq T$ for all $i < T^{+}$ . Let $A = \bigcup _{i < T^{+}} A_{i}$ , and suppose further that we are given some tuple d such that for all $i < T^{+}$ . By induction on $i < T^{+}$ we will build increasing and continuous sequences $\langle A^{\prime }_{i} : i < T^{+} \rangle $ and $\langle M_{i} : i < T^{+} \rangle $ satisfying the following for all $i < T^{+}$ :

(1) $A^{\prime }_{0} = A_{0}$ and $A^{\prime }_{\leq i} \equiv A_{\leq i}$ .

(2) $M_{i}\models T$ with $M_{i} = T$ and $A^{\prime }_{i} \subseteq M_{i}$ .

(3) .
To begin, we define $A^{\prime }_{0} = A_{0}$ and take $M_{0}$ be any model containing $A^{\prime }_{0}$ of size $T$ . Given $A^{\prime }_{\leq i}$ and $M_{\leq i}$ satisfying the requirements, we pick $A^{\prime \prime }_{i+1}$ such that $A^{\prime }_{\leq i} A^{\prime \prime }_{i+1} \equiv A_{\leq i+1}$ . Then we apply extension, to obtain $A^{\prime }_{i+1} \equiv _{A^{\prime }_{\leq i}} A^{\prime \prime }_{i+1}$ such that . Note that $A^{\prime }_{\leq i+1} \equiv A_{\leq i+1}$ . We define $M_{i+1}$ to be any model containing $A^{\prime }_{i+1}M_{i}$ of size $T$ . This satisfies the requirements.
At limit $\delta $ , we define $A^{\prime }_{\delta } = \bigcup _{i < \delta } A^{\prime }_{i}$ and $M_{\delta } = \bigcup _{i < \delta } M_{i}$ . This clearly satisfies (1) and (2) and (3) is trivial. Therefore this completes the construction.
Let $M = \bigcup _{i < T^{+}} M_{i}$ . Choose $d'$ such that $d\langle A_{i} : i < T^{+} \rangle \equiv d' \langle A^{\prime }_{i} : i < T^{+} \rangle $ , which is possible by (1). Then we have for all $i < T^{+}$ .
Towards contradiction, suppose that there is some $i < T^{+}$ with the property that . Then, in particular, we have . Additionally, because , we know, by symmetry and transitivity, that . By symmetry once more, we get , a contradiction. This shows that for all $i < T^{+}$ , completing the proof of this direction.
(2) $\implies $ (3) Suppose (3) fails, i.e., we are given A of size $\kappa ^{+}$ , $p \in S(A)$ , and an increasing continuous sequence of sets $\langle A_{i} : i < \omega \rangle $ such that $A_{i} \leq \kappa $ and p Kimdivides over $A_{i}$ for all $i < \kappa ^{+}$ . We will define an increasing continuous sequence of ordinals $\langle \alpha _{i} : i < \kappa ^{+} \rangle $ such that $\alpha _{i} \in \kappa ^{+}$ and $p \upharpoonright A_{\alpha _{i+1}}$ Kimdivides over $A_{\alpha _{i}}$ for all $i < \kappa ^{+}$ . We set $\alpha _{0} = 0$ and given $\langle \alpha _{j} : j \leq i \rangle $ , we know that there is some formula $\varphi (x;a_{i+1}) \in p$ that Kimdivides over $A_{\alpha _{i}}$ , by our assumption on p. Let $\alpha _{i+1}$ be the least ordinal $< \kappa ^{+}$ such that $a_{i+1}$ is contained in $A_{\alpha _{i+1}}$ . For limit i, if we are given $\langle \alpha _{j} : j < i \rangle $ , we put $\alpha _{i} = \sup _{j < i} \alpha _{j}$ . Then we define $\langle A_{i} : i < \kappa ^{+} \rangle $ by $A^{\prime }_{i} = A_{\alpha _{i}}$ for all $i < \kappa $ , and let $d \models p$ be any realization. By construction, we have for all $i < \kappa ^{+}$ , which witnesses the failure of (2).
(3) $\implies $ (1) This was established in [Reference Kaplan, Ramsey and Shelah9, Theorem 3.9].
Remark 3.7. In [Reference Casanovas and Kim2, Proposition 4.6] it is shown that in every theory with TP $_{2}$ , there is an increasing chain of sets $\langle D_{i} : i < T^{+} \rangle $ and tuple d such that $D_{i} \leq T$ and for all $i < T^{+}$ . Hence, for nonsimple NSOP $_{1}$ theories, the condition of continuity in the statement of Proposition 3.6 is essential.
The following theorem will be referred to as ‘witnessing.’ It shows that Morley sequences are witnesses to Kimdividing. Over models this was established in [Reference Kaplan and Ramsey8, Theorem 5.1], however for us it will be deduced as a corollary of Proposition 3.6.
Theorem 3.8. Suppose T is NSOP $_{1}$ with existence and $I = \langle a_{i} : i < \omega \rangle $ is an Morley sequence over A. If $\varphi (x;a_{0})$ Kimdivides over A, then $\{\varphi (x;a_{i}) : i < \omega \}$ is inconsistent.
Proof Suppose towards contradiction that $\varphi (x;a_{0})$ Kimdivides over A and $I = \langle a_{i} : i < \omega \rangle $ is an Morley sequence over A such that $\{\varphi (x;a_{i}) : i < \omega \}$ is consistent. By naming A as constants, we may assume $A \leq T$ . We may stretch I such that $I = \langle a_{i} : i < T^{+} \rangle $ . Define $A_{i} = Aa_{<i}$ . Then $\langle A_{i} : i < T^{+} \rangle $ is increasing and continuous and $A_{i} \leq T$ . Let $d \models \{\varphi (x;a_{i}) : i < T^{+} \}$ . We claim for all $i < T^{+}$ . If not, then for some $i < T^{+}$ , we have , or, in other words . Since I is an Morley sequence, we also have , hence , by transitivity (Theorem 3.4). This entails, in particular, that , which is a contradiction, since $\varphi (x;a_{i})$ Kimdivides over A. This completes the proof.
4 Low NSOP $_{1}$ theories
This section is dedicated to proving that Lascar and Shelah strong types coincide in any low NSOP $_{1}$ theory with existence. This generalizes the corresponding result of Buechler for low simple theories [Reference Buechler1] (also independently discovered by Shami [Reference Shami17]).
Definition 4.1. We say that the theory T is low if, for every formula $\varphi (x;y)$ , there is some $k < \omega $ , such that if $I = \langle a_{i} : i < \omega \rangle $ is an indiscernible sequence and $\{\varphi (x;a_{i}) : i < \omega \}$ is inconsistent, then it is kinconsistent.
In [Reference Buechler1], the definition of lowness is given in terms of the finiteness of certain $D(p,\varphi )$ ranks, which we will not need here. However, as observed in [Reference Buechler1], the above definition coincides with this definition in the case that T is simple.
Lemma 4.2. Suppose T is NSOP $_{1}$ with existence. Assume we are given tuples $(a_{i})_{i \leq n}$ and $L(A)$ formulas $(\varphi _{i}(x;y_{i}))_{i \leq n}$ such that for all $i \leq n$ . Then the following are equivalent:

(1) The formula $\bigwedge _{i \leq n} \varphi _{i}(x;a_{i})$ does not Kimdivide over A.

(2) For all Aindiscernible sequences $\langle \overline {a}_{j} : j < \omega \rangle = \langle (a_{j,0},\ldots , a_{j,n}) : j < \omega \rangle $ with $(a_{0,0},\ldots , a_{0,n}) = (a_{0},\ldots , a_{n})$ and for all $j < \omega $ and $i \leq n$ , the following set of formulas does not Kimdivide over A:
$$ \begin{align*} \left\{ \bigwedge_{i \leq n} \varphi_{i}(x;a_{j,i}) : j < \omega \right\}. \end{align*} $$ 
(3) There is an Aindiscernible sequence $\langle \overline {a}_{j} : j < \omega \rangle = \langle (a_{j,0},\ldots , a_{j,n}) : j < \omega \rangle $ with $(a_{0,0},\ldots , a_{0,n}) = (a_{0},\ldots , a_{n})$ and for all $j < \omega $ , $i \leq n$ such that
$$ \begin{align*} \left\{ \bigwedge_{i \leq n} \varphi_{i}(x;a_{j,i}) : j < \omega \right\} \end{align*} $$is consistent.
Proof (1) $\implies $ (2) Suppose we are given $\langle \overline {a}_{j} : j < \omega \rangle $ as in (2) and let $c \models \bigwedge _{i \leq n} \varphi _{i}(x;a_{i})$ with . As $\langle \overline {a}_{j} : j < \omega \rangle $ is Aindiscernible, for each $j> 0$ , there is some $\sigma _{j} \in \mathrm {Autf}(\mathbb {M}/A)$ with