Transitivity of Kim-independence

TL;DR

This paper establishes that Kim-independence satisfies transitivity in NSOP$_{1}$ theories and characterizes witnesses to Kim-dividing, providing new insights and resolving open questions in the model theory of NSOP$_{1}$ theories.

Contribution

It proves transitivity of Kim-independence in NSOP$_{1}$ theories and characterizes Kim-dividing witnesses, advancing understanding of independence in these theories.

Findings

Kim-independence satisfies transitivity in NSOP$_{1}$ theories.

Witnesses to Kim-dividing are exactly the $ ext{ind}^K$-Morley sequences.

Several open questions on transitivity and Morley sequences are answered.

Abstract

We prove several results on the behavior of Kim-independence upon changing the base in NSOP$_{1}$ theories. As a consequence, we prove that Kim-independence satisfies transitivity and that this characterizes NSOP$_{1}$. Moreover, we characterize witnesses to Kim-dividing as exactly the $\ind^{K}$-Morley sequences. We give several applications, answering a number of open questions concerning transitivity, Morley sequences, and local character in NSOP$_{1}$ theories.

Equations

$$\text{Av}(\mathcal{D},B)=\{\phi (x;b):\{a\in A:\mathbb{M}\models \phi (a,b)\}\in \mathcal{D}\}.$$

$${\mathcal{T}}_{\alpha }\upharpoonright w=\{\eta \in {\mathcal{T}}_{\alpha }:\min (\text{dom}(\eta ))\in w\text{ and }\beta \in \text{dom}(\eta )\setminus w⟹\eta (\beta )=0\}.$$

$${\text{qftp}}_{L^{\prime }}(s_0,\dots ,s_{n-1})={\text{qftp}}_{L^{\prime }}(t_0,\dots ,t_{n-1}),$$

$$\text{tp}(a_{s_0},\dots ,a_{s_{n-1}}/A)=\text{tp}(a_{t_0},\dots ,a_{t_{n-1}}/A).$$

$${\text{qftp}}_{L^{\prime }}(t_0,\dots ,t_{n-1})={\text{qftp}}_{L^{\prime }}(s_0,\dots ,s_{n-1})$$

$${\text{tp}}_{\Delta }(b_{t_0},\dots ,b_{t_{n-1}})={\text{tp}}_{\Delta }(a_{s_0},\dots ,a_{s_{n-1}}).$$

$$(a_{\eta }{)}_{\eta \in {\mathcal{T}}_{\alpha }\upharpoonright w}{\equiv }_C(a_{\eta }{)}_{\eta \in {\mathcal{T}}_{\alpha }\upharpoonright v}$$

$$(a_{\eta }{)}_{\eta \in {\mathcal{T}}_{\kappa }\upharpoonright v}{\equiv }_M(b_{\eta }{)}_{\eta \in {\mathcal{T}}_{\omega }\upharpoonright w}.$$

$$\Gamma (x;N,b)=\text{tp}(c/N)\cup \{\neg \phi (x,b;n):\phi (x,y;n)\text{ Kim-divides over }M\}.$$

$$c_n\models \bigcup _{i<n}\Gamma (x;N,b_i).$$

$$\Delta (x;N,b)$$

$$\Gamma (x;N,b)⊢\psi (x;b),$$

$$\bigcup _{i<\omega }\Gamma (x;N,b_i)⊢\{\psi (x;b_i):i<\omega \}.$$

$${\Gamma }_{\alpha ,i+1}({\overline{x}}_{\alpha ,i+1})={\Gamma }_{\alpha ,i}({\overline{x}}_{\alpha ,i})\cup \{(\exists y)\phi (y;\overline{x})\to \phi (x_{\lambda };\overline{x}):\lambda =\phi (y;\overline{x})\in {\Lambda }_i\}.$$

$$\Delta (\overline{x})=\Gamma (\overline{x})\cup \{\neg \phi ({\overline{x}}_{\alpha };a_{\alpha }):\phi ({\overline{x}}_{\alpha },y)\in L(M),\phi ({\overline{x}}_{\alpha };a)\text{ Kim-divides over }M,\alpha <{\kappa }^{+}\},$$

$$(a_{n\cdot i},a_{n\cdot i+1},\ldots,a_{n\cdot i+(n-1)})\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{$\scriptscriptstyle\textnormal{}$\hss}\,\,\,\,}}^{K}_{M}(a_{n\cdot j},a_{n\cdot j+1},\ldots,a_{n\cdot j+(n-1)})_{j<i},$$

$$(a_{\eta }{)}_{\eta \in {\mathcal{T}}_{\kappa }\upharpoonright v}{\equiv }_M(b_{\eta }{)}_{\eta \in {\mathcal{T}}_{\omega }\upharpoonright w}.$$

$$(b_{\eta }{)}_{\eta \in {\mathcal{T}}_{\omega }\upharpoonright w}{\equiv }_M(b_{\eta }{)}_{\eta \in {\mathcal{T}}_{\omega }\upharpoonright v}.$$

$$(a_{\eta }{)}_{\eta \in {\mathcal{T}}_{\alpha }\upharpoonright w}{\equiv }_M(a_{\eta }{)}_{\eta \in {\mathcal{T}}_{\alpha }\upharpoonright v}$$

$$J\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{$\scriptscriptstyle\textnormal{}$\hss}\,\,\,\,}}^{K}_{M}N,$$

$$I^{\prime}_{\alpha,>\alpha}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{$\scriptscriptstyle\textnormal{}$\hss}\,\,\,\,}}^{K}_{M}J.$$

$$N^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{$\scriptscriptstyle\textnormal{}$\hss}\,\,\,\,}}^{K}_{M}I^{\prime}_{\alpha,>\alpha}.$$

$$(b_{\eta {|}_{[\beta ,\alpha )},i}^{\alpha }{)}_{\beta \in \alpha \setminus \text{lim}(\alpha )}{I}_{\alpha ,>\alpha }^{\prime \prime }{\equiv }_M(b_{\eta {|}_{[\beta ,\alpha )},i}^{\alpha }{)}_{\beta \in \alpha \setminus \text{lim}(\alpha )}{I}_{\alpha ,>\alpha }^{\prime }{\equiv }_{M}I.$$

$$(c_{{\eta }^{\prime }{|}_{[\beta ,\alpha +1)}}^{\prime }{)}_{\beta \in (\alpha +1)\setminus \text{lim}(\alpha +1)}{I}_{\alpha ,>\alpha +1}^{\prime \prime \prime }{\equiv }_M(c_{{\eta }^{\prime }{|}_{[\beta ,\alpha +1)}}{)}_{\beta \in (\alpha +1)\setminus \text{lim}(\alpha +1)}{I}_{\alpha ,>\alpha +1}^{\prime \prime }{\equiv }_{M}I,$$

$$(b_{\eta }^{\kappa }{)}_{\eta \in {\mathcal{T}}_{\kappa }\upharpoonright v}{\equiv }_N(c_{\eta }{)}_{\eta \in {\mathcal{T}}_{\omega }\upharpoonright w},$$

$$(c_{\eta }{)}_{\eta \in {\mathcal{T}}_{\omega }\upharpoonright w}{\equiv }_N(c_{\eta }{)}_{\eta \in {\mathcal{T}}_{\omega }\upharpoonright v}.$$

$$\bigcup _{i<\omega }\Delta (x_0,\dots ,x_K,b_{K+1,i}^{\prime })$$

$$\mathbb{M}\models \phi (a_{<n})⟺\{i\in \alpha :\mathbb{M}\models \phi (b_{i,<n})\}\in \mathcal{U}.$$

Full text

The first author would like to thank the Israel Science Foundation for partial support of this research (grants No. 1533/14 and 1254/18).

1. 1 Introduction
2. 2 Preliminaries
3. 3 Transitivity holds in NSOP1 theories
4. 4 Transitivity implies NSOP1
5. 5 \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequences are witnesses
6. 6 Applications

1. Introduction

The class of NSOP1 theories may be viewed as the class of theories that are simple at a generic scale. This picture emerged piecemeal, starting with the results of Chernikov and the second-named author [CR16], which established a Kim-Pillay-style criterion for NSOP1 and characterized the NSOP1 theories in terms of a weak variant of the independence theorem. Simplicity-like behavior had been observed in certain algebraic structures—for example, the generic vector space with a bilinear form studied by Granger, and $\omega$-free PAC fields investigated by Chatzidakis—and these new results established these structures are NSOP1 and suggested that this simplicity-like behavior might be characteristic of the class. The analogy with simplicity theory was deepened in [KR20] and [KRS19] with the introduction of Kim-independence. There it was shown that, in an NSOP1 theory, Kim-independence satisfies appropriate versions of Kim’s lemma, symmetry, the independence theorem, and local character and that, moreover, these properties individually characterize NSOP1 theories. This notion of independence has proved useful in proving preservation of NSOP1 under various model-theoretic constructions and has been shown to coincide with natural algebraic notions of independence in new concrete examples. In this way, the structure theory for NSOP1 theories has developed along parallel lines to simplicity theory, with Kim-independence replacing the core notion of non-forking independence.

The key difference between these settings stems from the fact that the notion of Kim-independence only speaks about the behavior of dividing at the generic scale. To say that $a$ is Kim-independent over $M$ with $b$ is to say that any $M$-indiscernible sequence $I$ beginning with $b$, if sufficiently generic over $M$, is conjugate over $Mb$ to one that is indiscernible over $Ma$. In the initial definition of Kim-independence, genericity is understood to mean that the sequence is a Morley sequence in a global $M$-invariant type, but, after the fact, it turns out that broader notions of generic sequence give rise to equivalent definitions in the context of NSOP1 theories [KR20, Theorem 7.7]. In any case, this additional genericity requirement in the definition of independence produces a curious phenomenon: roughly speaking, asserting indiscernibility over a larger base is making a stronger statement, asserting genericity over a bigger base is making a weaker one. This tension is what introduces subtleties in the generalization of facts from non-forking independence in simple theories to the broader setting of Kim-independence in NSOP1 theories, as base monotonicity no longer holds. In fact, an NSOP1 theory in which Kim-independence satisfies base monotonicity is necessarily simple [KR20, Proposition 8.8].

This paper is devoted to studying the ways that genericity over one base may be transfered to genericity over another base. Base monotonicity trivializes all such questions in the context of non-forking independence in simple theories, so the issues we deal with here are new and unique to the NSOP1 world. The first work along these lines was in [KR18], where Kruckman and the second-named author proved “algebraically reasonable” versions of extension, the independence theorem, and the chain condition, which allow one to arrange for tuples to be Kim-independent over a given base and algebraically independent over a larger one. We build on this work, showing that in many cases one can arrange for Kim-independence over both bases and extend this to the construction of Morley sequences. This leads to our main theorem:

Theorem**.**

Suppose $T$ is a complete theory. The following are equivalent:

1. (1)

$T$* is NSOP1* 2. (2)

Transitivity of Kim-independence over models: if $M\prec N\models T$, a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N and a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}b, then a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Nb.111In the literature, transitivity for a relation \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}} is sometimes taken to mean a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{A}b+a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{Ab}c\iff a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{A}bc, which implies base monotonicity. Since, in general, \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K} does not satisfy base monotonicity in an NSOP1 theory, we use transitivity to denote only the $\implies$ direction. This is reasonable since this may be paraphrased by saying that a non-Kim-forking extension of a non-Kim-forking extension is a non-Kim-forking extension (all extensions over models). Kim has suggested using the term “transitivity lifting” for this notion, but we opt for the simpler “transitivity.”** 3. (3)

\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequences over models are witnesses: if $M\models T$ and $\varphi(x;b_{0})$ Kim-divides over $M$ and $\langle b_{i}:i<\omega\rangle$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence, then $\{\varphi(x;b_{i}):i<\omega\}$ is inconsistent.

The direction $(3)\implies(1)$ was known by [KR20, Theorem 3.16], but all other directions are new. We prove $(1)\implies(2)$ in Theorem 3.4, $(2)\implies(1)$ in Proposition 4.3, and finally $(1)\implies(3)$ in Theorem 5.1 below.

The theorem clarifies the extent to which concepts from simplicity theory can be carried over to the NSOP1 context. The Kim-Pillay theorem for simple theories catalogues the basic properties of non-forking independence in a simple theory. We had showed all of these properties for Kim-independence except base monotonicity, transitivity, and local character in [KR20], and observed that base monotonicity had to go for non-simple NSOP1 theories. Local character was later established in joint work with Shelah in [KRS19], which left only transitivity. An alternative formulation of transitivity, which is a consequence of the standard one and base monotonicity, was considered in [KR20, Section 9.2], where it was shown to fail in NSOP1 theories in general. The present theorem establishes transitivity in its usual form and, moreover, goes further, showing that transitivity of Kim-independence is characteristic of NSOP1 theories.

This theorem also represents a signficant technical development in the study of Kim-independence, allowing us to answer several questions. The (1)$\implies$(2) direction and its proof settle two questions from our prior work [KRS19, Question 3.14, Question 3.16]. The (1)$\implies$(3) direction collapses two kinds of generic sequence studied in [KR20]: it has as a corollary that tree Morley sequences coincide with total \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequences, answering [KR20, Question 7.12] and, additionally, gives a characterization of witnesses for Kim-dividing in NSOP1 theories.

We give three applications in Section 6. First, we prove two ‘lifting lemmas’ that show that, in an NSOP1 theory, if $M$ is an elementary substructure of $N$, then whenever a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N, all \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequences and tree Morley sequences over $M$ beginning with $a$ are conjugate over $Ma$ to sequences that are respectively \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley or tree Morley over $N$. This gives an analogue to a known result for non-forking Morley sequences in simple theories and clarifies the relationship between witnesses to Kim-dividing between two bases, one contained in another. Secondly, we prove a local version of preservation of Kim-independence under unions of chains, which was previously only known for complete types. In an NSOP1 theory, a formula $k$-Kim-divides over an increasing union of models if and only if it $k$-Kim-divides over a cofinal collection of models in the chain (for an appropropriate definition of $k$-Kim-dividing), which answers [KRS19, Question 3.17]. Finally, we reformulate the Kim-Pillay-style characterization of \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K} from [KRS19, Theorem 9.1], instead characterizing \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K} intrinsically in terms of properties of an abstract independence relation, without reference to finite satisfiability. We expect that these results will have further applications in the study of this class of theories.

2. Preliminaries

Throughout the paper, $T$ will denote a complete theory in the language $L$ with infinite monster model $\mathbb{M}\models T$. We will not notationally distinguish between elements and tuples. We will write $x,y,$ and $z$ to denote tuples of variables, and use the letters $M,N$ to denote models of $T$.

2.1. NSOP1 theories, invariant types, and Morley sequences

Definition 2.1**.**

[DS04, Definition 2.2] A formula $\varphi\left(x;y\right)$ has the $1$-strong order property (SOP${}_{1})$ if there is a tree of tuples $(a_{\eta})_{\eta\in 2^{<\omega}}$ so that

• •

For all $\eta\in 2^{\omega}$, the partial type $\{\varphi\left(x;a_{\eta\restriction n}\right):n<\omega\}$ is consistent.

• •

For all $\nu,\eta\in 2^{<\omega}$, if $\nu\frown\langle 0\rangle\unlhd\eta$ then $\left\{\varphi\left(x;a_{\eta}\right),\varphi\left(x;a_{\nu\frown\langle 1\rangle}\right)\right\}$ is inconsistent.

A theory $T$ is NSOP**1 if no formula has SOP1 modulo $T$.

The following equivalent formulation is more useful in practice:

Fact 2.2**.**

[CR16, Lemma 5.1] [KR20, Proposition 2.4] A theory $T$ has NSOP1 if and only if there is a formula $\varphi\left(x;y\right)$, $k<\omega$, and an infinite sequence $\langle\overline{c}_{i}:i\in I\rangle$ with $\overline{c}_{i}=\left(c_{i,0},c_{i,1}\right)$ satisfying:

1. (1)

For all $i\in I$, $c_{i,0}\equiv_{\overline{c}_{<i}}c_{i,1}$. 2. (2)

$\{\varphi\left(x;c_{i,0}\right):i\in I\}$ is consistent. 3. (3)

$\{\varphi\left(x;c_{i,1}\right):i\in I\}$ is $k$-inconsistent.

Moreover, if $T$ has SOP1, there is such a $\varphi$ with $k=2$.

Given an ultrafilter $\mathcal{D}$ on a set of tuples $A$, we may define a complete type $\text{Av}(\mathcal{D},B)$ over $B$ by

[TABLE]

We write a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}^{u}B to mean $\text{tp}\left(a/MB\right)$ is finitely satisfiable in $M$, in other words $\text{tp}(a/MB)$ is a coheir of its restriction to $M$. This is additionally equivalent to asserting that there is an ultrafilter $\mathcal{D}$ on tuples from $M$ such that $a\models\text{Av}(\mathcal{D},MB)$. A global type $q\in S\left(\mathbb{M}\right)$ is called $A$-invariant if $b\equiv_{A}b^{\prime}$ implies that, for all $\varphi(x;y)$, we have $\varphi\left(x;b\right)\in q$ if and only if $\varphi\left(x;b^{\prime}\right)\in q$. A global type $q$ is invariant if there is some small set $A$ such that $q$ is $A$-invariant. If $M$ is a model, then any type $p\in S(M)$ is finitely satisfiable in $M$ and hence $p=\text{Av}(\mathcal{D},M)$ for some ultrafilter $\mathcal{D}$ on tuples from $M$. Then $\text{Av}(\mathcal{D},\mathbb{M})$ is a global $M$-finitely satisfiable (and hence $M$-invariant) extension of $p$ (see, e.g., [She90, Lemma VII.4.1]).

Definition 2.3**.**

Suppose $q$ is an $A$-invariant global type and $I$ is a linearly ordered set. By a *Morley sequence in *$q$ over $A$ of order type $I$, we mean a sequence $\langle b_{\alpha}:\alpha\in I\rangle$ such that for each $\alpha\in I$, $b_{\alpha}\models q|_{Ab_{<\alpha}}$ where $b_{<\alpha}=\langle b_{\beta}:\beta<\alpha\rangle$. Given a linear order $I$, we will write $q^{\otimes I}$ for the unique global $A$-invariant type in variables $\langle x_{\alpha}:\alpha\in I\rangle$ such that for any $B\supseteq A$, if $\overline{b}\models q^{\otimes I}|_{B}$ then $b_{\alpha}\models q|_{Bb_{<\alpha}}$ for all $\alpha\in I$. If $q$ is, moreover, finitely satisfiable in $A$, in which case b_{\alpha}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{A}b_{<\alpha} for all $\alpha\in I$, then we refer to a Morley sequence in $q$ over $A$ as a coheir sequence over $A$.

We will also make use of the dual notions of heir and an heir sequence:

Definition 2.4**.**

If $B\supseteq M$, we say that $p\in S(B)$ is an heir of its restriction to $M$ if B\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{M}a for some, equivalently all, $a\models p$ and we write a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{h}_{M}b if and only if $\text{tp}(a/Mb)$ is an heir of $\text{tp}(a/M)$ if and only if b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{M}a. We say that $\langle b_{i}:i\in I\rangle$ is an indiscernible heir sequence over $M$ if $\langle b_{i}:i\in I\rangle$ is $M$-indiscernible and b_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{h}_{M}b_{<i} for all $i\in I$.

Definition 2.5**.**

Suppose $M$ is a model.

1. (1)

We say that $\varphi\left(x;b\right)$ *Kim-divides over *$M$ if there is a global $M$-invariant $q\supseteq\text{tp}\left(b/M\right)$ and Morley sequence $\langle b_{i}:i<\omega\rangle$ over $M$ in $q$ with $\{\varphi(x;b_{i}):i<\omega\}$ inconsistent. 2. (2)

We say that $\varphi\left(x;b\right)$ Kim-forks over $M$ if it implies a finite disjunction of formulas, each Kim-dividing over $M$. 3. (3)

A type $p$ Kim-forks over $M$ if there is $\varphi(x;b)$ such that $p\vdash\varphi(x;b)$ and $\varphi(x;b)$ Kim-forks over $M$. 4. (4)

We write a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}^{K}B for $\text{tp}\left(a/MB\right)$ does not Kim-fork over $M$. We may also paraphrase a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}B as $a$ and $B$ are Kim-independent over $M$. 5. (5)

We say that an infinite sequence $\langle a_{i}:i\in I\rangle$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$ if $\langle a_{i}:i\in I\rangle$ is $M$-indiscernible and a_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}a_{<i} for all $i\in I$.

Note that if a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}^{u}B then a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}^{f}B (i.e. $\text{tp}\left(a/BM\right)$ does not fork over $M$) which implies a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}^{K}B.

Kim-independence may be used to give several equivalents of NSOP1. In order to state the appropriate form of local character for this notion, we will need to introduce the generalized club filter.

Definition 2.6**.**

Let $\kappa$ be a cardinal and $X$ a set with $\left|X\right|\geq\kappa$. We write $\left[X\right]^{\kappa}$ to denote $\{Y\subseteq X:\left|Y\right|=\kappa\}$ and likewise $[X]^{<\kappa}$ for $\bigcup_{\lambda<\kappa}[X]^{\lambda}$. A set $C\subseteq\left[X\right]^{\kappa}$ is club if, for every $Y\in\left[X\right]^{\kappa}$, there is some $Z\in C$ with $Y\subseteq Z$ and if, whenever $\langle Y_{i}:i<\alpha\leq\kappa\rangle$ is an increasing chain in $C$, i.e. each $Y_{i}\in C$ and $i<j<\alpha$ implies $Y_{i}\subseteq Y_{j}$, then $\bigcup_{i<\alpha}Y_{i}\in C$.

Fact 2.7**.**

[KR20, Theorem 8.1] [KRS19, Theorem 1.1] The following are equivalent for the complete theory $T$:

1. (1)

$T$ is NSOP1. 2. (2)

Kim’s lemma for Kim-dividing: Given any model $M\models T$ and formula $\varphi\left(x;b\right)$, $\varphi\left(x;b\right)$ Kim-divides if and only if for any $\langle b_{i}:i<\omega\rangle$ Morley over $M$ in some global $M$-invariant type, $\{\varphi\left(x;b_{i}\right):i<\omega\}$ is inconsistent. 3. (3)

Symmetry of Kim independence over models: a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}^{K}b iff b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}^{K}a for any $M\models T$. 4. (4)

Local character on a club: given any model $M\models T$ and type $p\in S(M)$, the set $\{N\prec M:|N|=|T|\text{ and }p\text{ does not Kim-divide over }N\}$ is a club subset of $[M]^{|T|}$. 5. (5)

Independence theorem over models: if A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}B, c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}A, c^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}B and $c\equiv_{M}c^{\prime}$ then there is some c^{\prime\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}AB such that $c^{\prime\prime}\equiv_{MA}c$ and $c^{\prime\prime}\equiv_{MB}c^{\prime}$.

Remark 2.8*.*

Because NSOP1 is preserved by naming constants, we also see that if $\kappa\geq|T|$ and we are given any model $M\models T$ with $|M|\geq\kappa$ and type $p\in S(M)$, the set $\{N\prec M:|N|=\kappa\text{ and }p\text{ does not Kim-divide over }N\}$ is a club subset of $[M]^{\kappa}$. This follows by choosing an arbitrary $M_{0}\prec M$ of size $\kappa$ and applying Fact 2.7(3) to the theory $T(M_{0})$ obtained from $T$ by adding constants for $M_{0}$.

We will make extensive use of the following additional properties of Kim-independence in NSOP1 theories:

Fact 2.9**.**

Suppose that $T$ is NSOP1 and $M\models T$.

1. (1)

Extension: if a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b, then given any $c$, there is $a^{\prime}\equiv_{Mb}a$ such that a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}bc [KR20, Proposition 3.20]. 2. (2)

Consistency along \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequences: suppose $\langle a_{i}:i<\omega\rangle$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$. Then if $\varphi\left(x,a_{0}\right)$ does not Kim-divide over $M$, then $\{\varphi\left(x,a_{i}\right):i<\omega\}$ does not Kim-divide over $M$, and in particular it is consistent [KR20, Lemma 7.6]. 3. (3)

Strengthened independence theorem: Suppose $c_{0}\equiv_{M}c_{1}$, c_{0}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}a, c_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b and a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b. Then there is $c\models\text{tp}(c_{0}/Ma)\cup\text{tp}(c_{1}/Mb)$ such that a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}bc, b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}ac, and c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}ab [KR18, Theorem 2.13].

We will need the following chain condition for \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequences, which is a slight elaboration of the proof of Fact 2.9(2).

Lemma 2.10**.**

Suppose $T$ is NSOP1 and $M\models T$. If a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b_{0} and $I=\langle b_{i}:i<\omega\rangle$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$, then there is $a^{\prime}\equiv_{Mb_{0}}a$ such that $I$ is $Ma^{\prime}$-indiscernible and a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}I.

Proof.

Let $p(x;b_{0})=\text{tp}(a/Mb_{0})$. By induction, we will choose $a_{n}$ such that $a_{n}\models\bigcup_{i\leq n}p(x;b_{i})$ and a_{n}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b_{\leq n}. For $n=0$, we put $a_{0}=a$. Given $a_{n}$, pick $a^{\prime}$ such that $a^{\prime}b_{n+1}\equiv_{M}ab_{0}$. Then, by invariance, we have a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b_{n+1} and, additionally, b_{n+1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b_{\leq n}, and a_{n}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b_{\leq n}. As $a^{\prime}\equiv_{M}a\equiv_{M}a_{n}$, we may apply the independence theorem to find $a_{n+1}$ such that $a_{n+1}\equiv_{Mb_{\leq n}}a_{n}$, $a_{n+1}\equiv_{Mb_{n+1}}a^{\prime}$, and a_{n+1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b_{\leq n+1}. In particular, $a_{n+1}\models\bigcup_{i\leq n+1}p(x;b_{i})$, completing the induction.

By compactness and finite character, we can find $a_{*}\models\bigcup_{i<\omega}p(x;b_{i})$ such that a_{*}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}I. By compactness, Ramsey, an automorphism, we may assume $I$ is $Ma_{*}$-indiscernible, completing the proof. ∎

2.2. Generalized indiscernibles and a class of trees

The construction of tree Morley sequences goes by way of an inductive construction of approximations to Morley trees indexed by a certain class of trees. Although the initial set-up is somewhat cumbersome, the definitions allow us to give simple and streamlined constructions. It will be convenient to use the notation and basic definitions that accompany the trees $\mathcal{T}_{\alpha}$ from [KR20, Section 5.1]. The subsection below consists entirely of this notation and these definitions which are reproduced for the readers’ convenience.

For an ordinal $\alpha$, let the language $L_{s,\alpha}$ be $\langle\unlhd,\wedge,<_{lex},(P_{\beta})_{\beta\leq\alpha}\rangle$. 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 $\beta$. Our trees will be understood to be an $L_{s,\alpha}$-structure for some appropriate $\alpha$. We recall the definition of a class of trees $\mathcal{T}_{\alpha}$ below:

Definition 2.11**.**

Suppose $\alpha$ is an ordinal. We define $\mathcal{T}_{\alpha}$ to be the set of functions $f$ such that

• •

$\text{dom}(f)$ is an end-segment of $\alpha$ of the form $[\beta,\alpha)$ for $\beta$ equal to [math] 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 non-empty (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 $\beta\leq\alpha$, $P_{\beta}=\{f\in\mathcal{T}_{\alpha}:\text{dom}(f)=[\beta,\alpha)\}$.

Remark 2.12*.*

Condition (1) in the definition of $\mathcal{T}_{\alpha}$ was stated incorrectly in the first arXiv version of [KR20] via the weaker requirement that $\text{dom}(f)$ is an end-segment, non-empty if $\alpha$ is limit. There, and below, the inductive constructions assume that $\mathcal{T}_{\alpha+1}$ consists of the empty function (the root) and countably many copies of $\mathcal{T}_{\alpha}$ given by $\{\langle i\rangle\frown\eta:i<\omega,\eta\in\mathcal{T}_{\alpha}\}$ (where this concatenation is defined below in Definition 2.13). But if $\alpha$ is a limit, this becomes false if we allow functions with domain $\{\alpha\}$ since the empty function is not an element of $\mathcal{T}_{\alpha}$ and therefore the function $\alpha\mapsto i$ is not of the form $\langle i\rangle\frown\eta$ for some $\eta\in\mathcal{T}_{\alpha}$. This is rectified by omitting functions whose domain is an end-segment of the form $[\beta,\alpha)$ for $\beta$ limit.

Definition 2.13**.**

Suppose $\alpha$ is an ordinal.

1. (1)

(Restriction) If $w\subseteq\alpha\setminus\text{lim}(\alpha)$, the restriction of $\mathcal{T}_{\alpha}$ *to the set of levels *$w$ is given by

[TABLE] 2. (2)

(Concatenation) If $\eta\in\mathcal{T}_{\alpha}$, $\text{dom}(\eta)=[\beta+1,\alpha)$ for some $\beta\in\alpha\setminus\text{lim}(\alpha)$, 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 $\eta\cup\{(\alpha,i)\}$. When we write $\langle i\rangle\in\mathcal{T}_{\alpha+1}$ by itself, we use this to denote the function $\{(\alpha,i)\}$. 3. (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. (4)

(The all [math]’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 $\beta\in\alpha\setminus\text{lim}(\alpha)$.

We will most often be interested in collections of tuples indexed by $\mathcal{T}_{\alpha}$ and, if $(a_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ is such a collection and $\eta\in\mathcal{T}_{\alpha}$, we will write $a_{\unrhd\eta}$ and $a_{\vartriangleright\eta}$ for tuples enumerating the elements indexed by elements of $\mathcal{T}_{\alpha}$ above or strictly above $\eta$ in the tree partial order, respectively. Note that if $\beta<\alpha$ is a limit ordinal and $\eta\in\mathcal{T}_{\alpha}$ has $\text{dom}(\eta)=[\beta+1,\alpha)$, then $\beta\frown\langle i\rangle$ is a function whose domain is $[\beta,\alpha)$ and is therefore not in $\mathcal{T}_{\alpha}$. If $(a_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ is a collection of tuples indexed by $\mathcal{T}_{\alpha}$, we will abuse notation and write $a_{\unrhd\eta\frown\langle i\rangle}$ for the tuple that enumerates $\{a_{\nu}:\nu\in\mathcal{T}_{\alpha},\eta\frown\langle i\rangle\subseteq\nu\}$ and likewise for $a_{\unrhd\zeta_{\beta}}$.

We additionally remark that the concatention notation is only unambiguous once we have specified in which tree the element lives—for example, $\langle i\rangle\frown\langle j\rangle$ can denote an element of $\mathcal{T}_{\alpha+2}$ when $\langle j\rangle\in\mathcal{T}_{\alpha+1}$ or an element of $\mathcal{T}_{\alpha+1}$ if $\langle i\rangle\in\mathcal{T}_{\alpha+1}$, but this notation reads unambiguously once we have specified in which tree we are referring to $\langle i\rangle\frown\langle j\rangle$. In the arguments below, the intended meaning of concatenation is clear from context and no confusion will arise.

The function $\iota_{\alpha\beta}$ includes $\mathcal{T}_{\alpha}$ into $\mathcal{T}_{\beta}$ by adding zeros to the bottom of every node in $\mathcal{T}_{\alpha}$. Clearly if $\alpha<\beta<\gamma$, then $\iota_{\alpha\gamma}=\iota_{\beta\gamma}\circ\iota_{\alpha\beta}$. If $\beta$ is a limit, then $\mathcal{T}_{\beta}$ is the direct limit of the $\mathcal{T}_{\alpha}$ for $\alpha<\beta$ along these maps.

Definition 2.14**.**

Suppose $I$ is an $L^{\prime}$-structure, where $L^{\prime}$ is some language.

1. (1)

We say that $(a_{i}:i\in I)$ is a set of $I$-indexed indiscernibles over $A$ if whenever

$(s_{0},\ldots,s_{n-1})$, $(t_{0},\ldots,t_{n-1})$ are tuples from $I$ with

[TABLE]

then we have

[TABLE] 2. (2)

In the case that $L^{\prime}=L_{s,\alpha}$ for some $\alpha$, we say that an $I$-indexed indiscernible is s-indiscernible. As the only $L_{s,\alpha}$-structures we will consider will be trees, we will often refer to $I$-indexed indiscernibles in this case as s-indiscernible trees. 3. (3)

We say that $I$-indexed indiscernibles have the modeling property if, given any $(a_{i}:i\in I)$ from $\mathbb{M}$ and set of parameters $A$, there is an $I$-indexed indiscernible $(b_{i}:i\in I)$ in $\mathbb{M}$ locally based on $(a_{i}:i\in I)$ over $A$ – i.e., given any finite set of formulas $\Delta$ from $L(A)$ and a finite tuple $(t_{0},\ldots,t_{n-1})$ from $I$, there is a tuple $(s_{0},\ldots,s_{n-1})$ from $I$ such that

[TABLE]

and also

[TABLE]

Recall that, given a set $X$, we write $[X]^{<\omega}$ to denote the set of finite subsets of $X$.

Definition 2.15**.**

Suppose $(a_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ is a tree of tuples, and $C$ is a set of parameters.

1. (1)

We say that $(a_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ is spread out over C if for all $\eta\in\mathcal{T}_{\alpha}$ with $\text{dom}(\eta)=[\beta+1,\alpha)$ for some $\beta\in\alpha$, there is a global $C$-invariant type $q_{\eta}\supseteq\text{tp}(a_{\unrhd\eta\frown\langle 0\rangle}/C)$ such that $(a_{\unrhd\eta\frown\langle i\rangle})_{i<\omega}$ is a Morley sequence over $C$ in $q_{\eta}$. 2. (2)

Suppose $(a_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ is a tree which is spread out and $s$-indiscernible over $C$ and for all $w,v\in[\alpha\setminus\text{lim}(\alpha)]^{<\omega}$ with $|w|=|v|$,

[TABLE]

then we say that $(a_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ is a Morley tree over $C$. 3. (3)

A tree Morley sequence over $C$ is a $C$-indiscernible sequence of the form $(a_{\zeta_{\beta}})_{\beta\in\alpha\setminus\text{lim}(\alpha)}$ for some Morley tree $(a_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ over $C$.

Remark 2.16*.*

Note that in Definition LABEL:spread_out_def(1), it is possible that $\mathrm{dom}(\eta)=[\beta+1,\alpha)$ for a limit ordinal $\beta\in\alpha$, in which case $\eta\frown\langle i\rangle$, defined to be the function $\eta\cup\{(\beta,i)\}$, is not an element of $\mathcal{T}_{\alpha}$. Nonetheless, the tuple $a_{\unrhd\eta\frown\langle i\rangle}$ still makes sense as the tuple whose elements are indexed by functions in the tree $\mathcal{T}_{\alpha}$ containing $\eta\frown\langle i\rangle$. See the remarks after Definition 2.13.

Fact 2.17**.**

1. (1)

For any $\alpha$, $\mathcal{T}_{\alpha}$-indexed indiscernibles have the modeling property [KKS14, Theorem 4.3] [KR20, Corollary 5.6]. 2. (2)

Given a model $M\models T$, there is a cardinal $\kappa$ such that if $(a_{\eta})_{\eta\in\mathcal{T}_{\kappa}}$ is a tree of tuples, spread out and $s$-indiscernible over $M$, then there is a Morley tree $(b_{\eta})_{\eta\in\mathcal{T}_{\omega}}$ such that for all $w\in[\omega]^{<\omega}$,

[TABLE]

for some $v\in[\kappa\setminus\text{lim}(\kappa)]^{<\omega}$ [KR20, Lemma 5.10].

The interest in tree Morley sequences is that the genericity condition is sufficiently weak that they exist under broader hypotheses than invariant Morley sequences, yet is sufficiently strong to witness Kim-independence. This is made precise below:

Definition 2.18**.**

Suppose $M$ is a model and $(a_{i})_{i<\omega}$ is an $M$-indiscernible sequence.

1. (1)

Say $(a_{i})_{i<\omega}$ is a witness for Kim-dividing over $M$ if, for all formulas $\varphi(x;a_{0})$ that Kim-divide over $M$, $\{\varphi(x;a_{i}):i<\omega\}$ is inconsistent. 2. (2)

Say $(a_{i})_{i<\omega}$ is a strong witness to Kim-dividing over $M$ if, for all $n$, the sequence $\langle(a_{n\cdot i},a_{n\cdot i+1},\ldots,a_{n\cdot i+n-1}):i<\omega\rangle$ is a witness to Kim-dividing over $M$.

Fact 2.19**.**

[KR20, Proposition 7.9] Suppose $T$ is NSOP1 and $M\models T$.

1. (1)

(Kim’s Lemma for tree Morley sequences) $\varphi(x;a)$ Kim-divides over $M$ if and only if $\{\varphi(x;a_{i}):i<\omega\}$ is inconsistent for some tree Morley sequence $(a_{i})_{i<\omega}$ over $M$ with $a_{0}=a$ if and only if $\{\varphi(x;a_{i}):i<\omega\}$ is inconsistent for all tree Morley sequences $(a_{i})_{i<\omega}$ over $M$ with $a_{0}=a$. [KR20, Corollary 5.14] 2. (2)

The sequence $(a_{i})_{i<\omega}$ is a strong witness for Kim-dividing over $M$ if and only if $(a_{i})_{i<\omega}$ is a tree Morley sequence over $M$. [KR20, Proposition 7.9]

Remark 2.20*.*

The argument for [KR20, Corollary 7.10] contains a proof that \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{f}-Morley sequences over models are strong witnesses to Kim-dividing. Note that it follows, then, that coheir sequences over models are also strong witnesses to Kim-dividing, as a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{M}b implies a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{f}_{M}b for all $M\models T$.

Finally, we define one more kind of \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence:

Definition 2.21**.**

Suppose $M\models T$. A total \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over$M$ is an $M$-indiscernible sequence $\langle a_{i}:i<\omega\rangle$ such that a_{>i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}a_{\leq i} for all $i<\omega$.

Fact 2.22**.**

Suppose $T$ is NSOP1 and $M\models T$.

1. (1)

If $I$ is a tree Morley sequence over $M$, then $I$ is a total \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$. 2. (2)

If $I$ is a total \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$, then $I$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley over $M$.

Proof.

(2) is obvious so we prove (1). Suppose $I=\langle a_{i}:i<\omega\rangle$ is a tree Morley sequence over $M$. Let $(a_{\eta})_{\eta\in\mathcal{T}_{\omega}}$ be a Morley tree over $M$ with $a_{i}=a_{\zeta_{i}}$ for all $i<\omega$. Then for all $i<\omega$, we have that $\langle a_{\unrhd\zeta_{i+1}\frown\langle j\rangle}:j<\omega\rangle$ is a Morley sequence over $M$ in a global $M$-invariant type which is $Ma_{\zeta_{\geq i+1}}$-indiscernible. Therefore a_{\zeta_{\geq i+1}}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}a_{\unrhd\zeta_{i+1}\frown\langle 0\rangle}, which, in particular, implies a_{>i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}a_{\leq i} for all $i<\omega$. ∎

3. Transitivity holds in NSOP1 theories

In this section, we prove the transitivity of Kim-independence in NSOP1 theories. The argument proceeds via an analysis of situations under which one can obtain sequences that are generic over more than one base simultaneously. The heart of the argument is Proposition 3.3, which proves the existence of a sequence that is a tree Morley sequence over a model and \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley over an elementary extension. This, combined with symmetry, gives transitivity as an immediate consequence.

Producing a sequence which is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley over a model and a tree Morley sequence over an elementary extension is less involved. The following lemma was implicit in [KRS19, Lemma 3.6]:

Lemma 3.1**.**

Suppose $T$ is NSOP1, $M\prec N\models T$, and a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N. Then there is a tree Morley sequence $(a_{i})_{i<\omega}$ over $N$ with $a_{0}=a$ such that a_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Na_{<i} for all $i<\omega$. In particular, $\langle a_{i}:i<\omega\rangle$ is simultaneously an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$ and a tree Morley sequence over $N$.

Proof.

Let $\langle a_{i}:i\in\mathbb{Z}\rangle$ be a coheir sequence over $N$ with $a_{0}=a$. Since coheir sequences are strong witnesses to Kim-dividing, by Remark 2.20, reversing the order of a sequence does not change whether or not it is a strong witness, and the fact that strong witnesses are tree Morley by Fact 2.19(2), it follows that, setting $b_{i}=a_{-i}$, we have that $\langle b_{i}:i<\omega\rangle$ is a tree Morley sequence over $N$ with $b_{0}=a$.

We claim this sequence also satisfies b_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Nb_{<i}: if not, then by symmetry, there is some $i$ such that Nb_{<i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}^{K}_{M}b_{i} and this is witnessed by some $\varphi(x,n;b_{i})\in\text{tp}(b_{<i}N/Mb_{i})$. Because $\langle a_{i}:i\in\mathbb{Z}\rangle$ is a coheir sequence over $N$, we have, in particular, that a_{>i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{N}a_{i}. Hence b_{<i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{N}b_{i} so there must be some $n^{\prime}\in N$ with $\models\varphi(n^{\prime},n;b_{i})$. But then N\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}^{K}_{M}b_{i}. By symmetry and invariance, this contradicts a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N, since $b_{i}\equiv_{N}a$. ∎

Lemma 3.2**.**

Supose $T$ is NSOP1 and $M\prec N\models T$. If b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N and c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N, then there is $c^{\prime}\equiv_{N}c$ such that bc^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N and b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}c^{\prime}.

Proof.

Define a partial type $\Gamma(x;N,b)$ by

[TABLE]

By Lemma 3.1, we may construct an $N$-indiscernible sequence $\langle b_{i}:i<\omega\rangle$ such that $b_{0}=b$, b_{i+1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Nb_{\leq i}, and $\langle b_{i}:i<\omega\rangle$ is a tree Morley sequence over $N$.

Claim 1: $\bigcup_{i<\omega}\Gamma(x;N,b_{i})$ is consistent.

Proof of claim: By induction on $n$, we will choose c_{n}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Nb_{<n} such that

[TABLE]

For $n=0$, we may set $c_{0}=c$ and the condition is satisfied since c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N.

Suppose we are given c_{n}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Nb_{<n} realizing $\bigcup_{i<n}\Gamma(x;N,b_{i})$. By extension, choose $c^{\prime}\equiv_{M}c$ with c^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b_{n}. As b_{n}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Nb_{<n}, we may apply the strengthened independence theorem, Fact 2.9(3), to find $c_{n+1}\models\text{tp}(c_{n}/Nb_{<n})\cup\text{tp}(c^{\prime}/Mb_{n})$ with c_{n+1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Nb_{<n+1} and b_{n}c_{n+1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Nb_{<n}. In particular, b_{n}c_{n+1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N, so $c_{n+1}\models\Gamma(x;N,b_{n})$. This gives $c_{n+1}\models\bigcup_{i<n+1}\Gamma(x;N,b_{i})$. The claim follows by compactness.∎

Now define a partial type $\Delta(x;N,b)$ by

[TABLE]

Claim 2: $\Delta(x;a,b)$ is consistent.

Proof of claim: Suppose not. Then, by definition of $\Delta(x;N,b)$, compactness, and the equality of Kim-forking and Kim-dividing, we have

[TABLE]

for some $\psi(x;b)\in L(Nb)$ that Kim-divides over $N$. Then we have

[TABLE]

The left-hand side is consistent by Claim 1 but the right hand side is inconsistent by Kim’s lemma and the choice of $\overline{b}$, a contradiction that proves the claim. ∎

Now let $c^{\prime}\models\Delta(x;N,b)$. Then, by symmetry, we have $c^{\prime}\equiv_{N}c$, bc^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N, and b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}c^{\prime} which is what we want. ∎

Proposition 3.3**.**

Suppose $T$ is NSOP1 and $M\prec N\models T$. If a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N, then there is a sequence $\langle a_{i}:i<\omega\rangle$ with $a_{0}=a$ which is a tree Morley sequence over $M$ and Kim-Morley over $N$.

Proof.

By induction on $\alpha$, we will construct trees $(a^{\alpha}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ satisfying the following conditions:

1. (1)

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

$(a^{\alpha}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ is $s$-indiscernible over $N$, spread out over $M$. 3. (3)

If $\alpha$ is a successor, then a^{\alpha}_{\emptyset}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}(a^{\alpha}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}. 4. (4)

(a^{\alpha}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N. 5. (5)

If $\alpha<\beta$, then for all $\eta\in\mathcal{T}_{\alpha}$, $a^{\beta}_{\iota_{\alpha\beta}(\eta)}=a^{\alpha}_{\eta}$.

To begin, put $a^{0}_{\emptyset}=a$. At a limit stage $\delta$, we define $(a^{\delta}_{\eta})_{\eta\in\mathcal{T}_{\delta}}$ by $a^{\delta}_{\iota_{\alpha\delta}(\eta)}=a^{\alpha}_{\eta}$ for all $\alpha<\delta$ and $\eta\in\mathcal{T}_{\alpha}$. This is well-defined by (5) and the definition of $\mathcal{T}_{\delta}$. Moreover, it clearly satisfies (1), (2) is trivial, and (3) and (4) are satisfied by finite character.

Now in the successor stage, we will construct $(a^{\alpha+1}_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}$. Let $\overline{b}=\langle(a^{\alpha}_{\eta,i})_{\eta\in\mathcal{T}_{\alpha}}:i<\omega\rangle$ be a coheir sequence over $M$ with $a^{\alpha}_{\eta,0}=a^{\alpha}_{\eta}$ for all $\eta\in\mathcal{T}_{\alpha}$. By (4), we may assume $\overline{b}$ is $N$-indiscernible and \overline{b}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N by the chain condition (Lemma 2.10). Apply Lemma 3.2 to get $b\equiv_{N}a$ such that b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}\overline{b} and b\overline{b}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N. Define a tree $(b_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}$ by $b_{\emptyset}=b$ and $b_{\langle i\rangle\frown\eta}=a^{\alpha}_{\eta,i}$ for all $i<\omega$, $\eta\in\mathcal{T}_{\alpha}$. Now let $(a^{\alpha+1}_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}$ be an $s$-indiscernible tree over $N$, locally based on the tree $(b_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ over $N$. By an automorphism, we may assume that $a^{\alpha+1}_{\langle 0\rangle\frown\eta}=a^{\alpha}_{\eta}$ for all $\eta\in\mathcal{T}_{\alpha}$, so (5) is satisfied. By construction and induction, $(b_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}$ is spread out over $N$ and $b_{\eta}\models\text{tp}(a/N)$ for all $\eta\in\mathcal{T}_{\alpha+1}$ so $(a^{\alpha+1}_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}$ satisfies (1) and (2). Likewise, since b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}\overline{b} and (b_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N by construction, and because of the fact that Kim-forking is witnessed by formulas, it follows from the fact that $(a^{\alpha+1}_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}$ is locally based on $(b_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}$ over $N$ that a^{\alpha+1}_{\emptyset}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}a^{\alpha+1}_{\vartriangleright\emptyset} and (a^{\alpha+1}_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N as well, so (3) and (4) are satisfied. This completes the construction.

Let $(a_{\eta})_{\eta\in\mathcal{T}_{\omega}}$ be the tree obtained by applying Fact 2.17. Then $(a_{\zeta_{\alpha}})_{\alpha<\omega}$ is the desired sequence. ∎

Theorem 3.4**.**

Suppose $T$ is NSOP1, $M\prec N\models T$, a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N and a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}b. Then a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Nb.

Proof.

Suppose $a,b,M,$ and $N$ are given as in the statement. By Proposition 3.3, there is a sequence $I=\langle a_{i}:i<\omega\rangle$ with $a_{0}=a$ such that $I$ is a tree Morley sequence over $M$ and an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $N$. Since b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}a, there is $I^{\prime}\equiv_{Na}I$ such that $I^{\prime}$ is $Nb$-indiscernible, by compactness, Ramsey, and Fact 2.19. Then $I^{\prime}$ still a tree Morley sequence over $M$ so, by Kim’s lemma for tree Morley sequences, Nb\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}a, so we may conclude by symmetry. ∎

Transitivity allows one to easily obtain analogues for Kim-independence of the “algebraically reasonable” properties of Kim- and algebraic-independence proved in [KR18]. For example, the following is the analogue of “algebraically reasonable extension” [KR18, Theorem 2.15]:

Corollary 3.5**.**

Suppose $T$ is NSOP1, $M\prec N\models T$, and a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N. Then given any $b$, there is $a^{\prime}$ with $a^{\prime}\equiv_{N}a$, a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Nb, and a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}b.

Proof.

Applying extension, we obtain $a^{\prime}\equiv_{N}a$ so that a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}b. By invariance, a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N so by transitivity, a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Nb. ∎

3.1. An example

In this subsection, we present an example that illustrates two important phenomena simultaneously. First, it shows that if $T$ is NSOP1, $M\prec N\models T$ and a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N, then it is not necessarily possible to find $I=\langle a_{i}:i<\omega\rangle$ that is a coheir sequence over $N$ with $a_{0}=a$ and I\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N. Our example shows that Lemma 3.1—this lemma shows that, in this situation, a coheir sequence starting with $a$ over $N$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley over $M$—is optimal, as, in general, one cannot improve it to conclude that a coheir sequence over $N$ is a stronger form of Morley sequence over $M$. In particular, it is not the case that every tree Morley sequence over $N$ is automatically a tree Morley sequence over $M$, as one might hope, as we produce a coheir sequence (which is therefore tree Morley over $N$) which is not tree Morley over $M$. Secondly, our example shows that it is possible, in an NSOP1 theory, that there is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence that is neither a tree Morley sequence nor a total \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence (later, in Corollary 5.5, we will show the notions of tree Morley sequence and total \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence are equivalent). In particular, we show that there is a sequence $\langle a_{i}:i<\omega\rangle$ that is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley over $M$ but a_{2}a_{3}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}^{K}_{M}a_{0}a_{1}.

Fact 3.6**.**

Let $L$ be the language consisting of a single binary function $f$.

1. (1)

The empty $L$-theory has a model completion $T_{f}$, which eliminates quantifiers. [Win75] [KR18, Corollary 3.10] 2. (2)

Modulo $T_{f}$, for all sets $A$, $\text{acl}(A)=\text{dcl}(A)=\langle A\rangle$, where $\langle A\rangle$ denotes the substructure generated by $A$. [KR18, Corollary 3.11] 3. (3)

$T_{f}$ is NSOP1 and Kim-independence coincides with algebraic independence: for any tuples $a,b$, if $M\models T_{f}$, a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b if and only if $\langle aM\rangle\cap\langle bM\rangle=M$. [KR18, Corollary 3.13]

Example 3.7**.**

Let $M$ be a countable model of $T_{f}$ and $N$ an $\aleph_{1}$-saturated elementary extension, all contained in the monster model $\mathbb{M}\models T_{f}$. Pick elements $m_{*}\in M$ and $n_{*}\in N\setminus M$. In $N\setminus M$, we can find a countable set of distinct elements $B=\{b_{i}:i<\omega\}$ such that $f|_{(B\times M)\cup(M\times B)\cup\Delta}=\{m_{*}\}$ and $f|_{(B\times B)\setminus\Delta}=\{n_{*}\}$, where $\Delta=\{(b_{i},b_{i}):i<\omega\}$. Let $\mathcal{D}$ be a non-principal ultrafilter on $N$ concentrating on $B$ and $q=\text{Av}(\mathcal{D},\mathbb{M})$. Let $I=(a_{i})_{i<\omega}\models q^{\otimes\omega}|_{N}$ be a Morley sequence in $q$ over $N$.

We claim a_{0}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N. This is equivalent to the assertion that $\langle a_{0}M\rangle\cap N=M$. Suppose $c\in\langle a_{0}M\rangle\cap N$. Then there is a term $t(x;m)$, possibly with parameters from $M$, such that $t(a_{0};m)=c$ and therefore $\{i:t(b_{i};m)=c\}\in\mathcal{D}$. One may easily check that if $s$ is a constant term in the language $L(Mb_{i})$, i.e. $L$ with constants for $M$ and $b_{i}$, then either $s=b_{i}$ or there is $m^{\prime}\in M$ with $s=m^{\prime}$. This is clear for the constants and, since $f(n,b_{i})=f(b_{i},n)=f(b_{i},b_{i})=m_{*}$ for all $n\in M$, the induction follows. Since the $b_{i}$ are pairwise distinct and $\{i:t(b_{i};m)=c\}\in\mathcal{D}$, it is clear that $c$ is not equal to any $b_{i}$, so it follows that $c\in M$.

However, $f(b_{i},b_{j})=n_{*}\in N\setminus M$ for all $i,j<\omega$ so $f(a_{0},a_{1})=n_{*}$. Therefore $\text{dcl}(a_{0},a_{1}M)\cap N\neq M$, which shows a_{0}a_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}^{K}_{M}N. This shows in particular I\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}^{K}_{M}N.

Next, in the proof of Lemma 3.1, we show that if $T$ is NSOP1, $M\prec N$ and b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N, then for any coheir sequence $\langle b_{i}:i<\omega\rangle$ in $\text{tp}(b/N)$, we have b_{>i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b_{i} for all $i<\omega$. It follows that a_{>i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}a_{i} and thus $I$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$ indexed in reverse. However, we have $f(a_{2},a_{3})=f(a_{0},a_{1})=n_{*}\not\in M$ so a_{0}a_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}^{K}_{M}a_{2}a_{3}, which shows that $I$ is not a total \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence.

4. Transitivity implies NSOP1

In this section, we complete the characterization of NSOP1 theories by the transitivity of Kim-independence. The argument is loosely inspired by the proof due to Kim that transitivity of non-forking independence implies simplicity [Kim01, Theorem 2.4]. However, we have to deal with a more complicated combinatorial configuration as well as the need to produce models over which we may observe a failure of transitivity from SOP1. We begin by observing a combinatorial consequence of SOP1 arising from the witnessing array of pairs and then work in a Skolemization of a given SOP1 theory to find the desired counter-example to transitivity.

Lemma 4.1**.**

Suppose $T$ has SOP1. Then there is a formula $\varphi(x;y)$ and an indiscernible sequence $(a_{i},c_{i,0},c_{i,1})_{i<\omega}$ such that

1. (1)

For all $i<\omega$, $a_{i}\models\{\varphi(x;c_{j,0}):j\leq i\}$. 2. (2)

$\{\varphi(x;c_{i,1}):i<\omega\}$* is $2$-inconsistent.* 3. (3)

For all $i<\omega$, $c_{i,0}\equiv_{a_{<i},c_{<i,0}c_{<i,1}}c_{i,1}$.

Proof.

Let $\kappa$ be a cardinal sufficiently large relative $|M|$ to apply Fact 2.17. Because $T$ has SOP1, we know by Fact 2.2 and compactness, there is a formula $\varphi(x;y)$ and an indiscernible sequence $(c_{i,0},c_{i,1})_{i<\kappa}$ such that

• •

$\{\varphi(x;c_{i,0}):i<\kappa\}$ is consistent.

• •

$\{\varphi(x;c_{i,1}):i<\kappa\}$ is $2$-inconsistent.

• •

For all $i<\kappa$, $c_{i,0}\equiv_{c_{<i,0}c_{<i,1}}c_{i,1}$ and $c_{n,0}\equiv_{a_{<n}c_{<n,0}c_{<n,1}}c_{n,1}$.

By induction on $n<\omega$, we will build $(a_{i})_{i<n}$ and $(c^{n}_{i,0},c^{n}_{i,1})_{i<\kappa}$ such that, for all $n<\omega$,

1. (1)

$\{\varphi(x;c^{n}_{i,0}):i<\kappa\}$ is consistent. 2. (2)

$\{\varphi(x;c^{n}_{i,1}):i<\kappa\}$ is $2$-inconsistent. 3. (3)

For all $i<\kappa$, $c^{n}_{i,0}\equiv_{c^{n}_{<i,0}c^{n}_{<i,1}}c^{n}_{i,1}$ and $c^{n}_{n,0}\equiv_{a_{<n}c^{n}_{<n,0}c^{n}_{<n,1}}c^{n}_{n,1}$. 4. (4)

For all $i<n$, $a_{i}\models\{\varphi(x;c^{n}_{j,0}):j\leq i\}$. 5. (5)

For $m\leq n$, $(c^{m}_{m,0},c^{m}_{m,1})=(c^{n}_{m,0},c^{n}_{m,1})$.

To begin, we define $(c^{0}_{i,0},c^{0}_{i,1})_{i<\kappa}$ by setting $(c^{0}_{i,0},c^{0}_{i,1})=(c_{i,0},c_{i,1})$ for all $i<\kappa$. This, together with the empty sequence of $a_{i}$’s satisfies (1)—(3). For $n=0$, (4) and (5) are vacuous, so this handles the base case.

Now suppose for $n$, we have constructed $(a_{i})_{i<n}$, $(c^{n}_{i,0},c^{n}_{i,1})_{i<\kappa}$. Choose $a_{n}\models\{\varphi(x;c^{n}_{i,0}):i\leq n\}$. Now by the pigeonhole principle, there are $i_{*}>j_{*}>n$ such that $c^{n}_{i_{*},1}\equiv_{a_{\leq n}c^{n}_{\leq n,0}c^{n}_{\leq n,1}}c^{n}_{j_{*},1}$. As $c^{n}_{i_{*},0}\equiv_{c^{n}_{<i_{*},0}c^{n}_{<i_{*},1}}c^{n}_{i_{*},1}$, there is $\sigma\in\text{Aut}(\mathbb{M}/c^{n}_{<i_{*},0}c^{n}_{<i_{*},1})$ with $\sigma(c^{n}_{i_{*},0})=c^{n}_{i_{*},1}$. Define a new array by setting $(c^{n+1}_{m,0},c^{n+1}_{m,1})=(c^{n}_{m,0},c^{n}_{m,1})$ for all $m\leq n$, $(c^{n+1}_{n+1,0},c^{n+1}_{n+1,1})=(c^{n}_{i_{*},1},c^{n}_{j_{*},1})$, and finally $(c^{n+1}_{n+\alpha,0},c^{n+1}_{n+\alpha,1})=\sigma(c^{n}_{i_{*}+\alpha,0},c^{n}_{i_{*}+\alpha,1})$ for all $2\leq\alpha<\kappa$.

Now we check that this satisfies the requirements. For (1), note that $\{\varphi(x;c^{n+1}_{i,0}):i<\kappa\}$ is equal to $\{\varphi(x;\sigma(c^{n}_{i,0})):i\leq n\text{ or }i\geq i_{*}\}$ and this is consistent because $\{\varphi(x;c^{n}_{i,0}):i\leq n\text{ or }i\geq i_{*}\}$ is consistent and $\sigma$ is an automorphism. Likewise, $\{\varphi(x;c^{n+1}_{i,1}):i<\kappa\}$ is equal to $\{\varphi(x;\sigma(c^{n}_{i,1})):i\leq n,i=j_{*},\text{ or }i>i_{*}\}$, so this is $2$-inconsistent because $\{\varphi(x;c^{n}_{i,1}):i<\kappa\}$ is $2$-inconsistent and $\sigma$ is an automorphism. (3)—(5) are immediate from our construction. This completes the induction.

Now define $(c^{\omega}_{i,0},c^{\omega}_{i,1})_{i<\omega}$ such that $(c^{\omega}_{i,0},c^{\omega}_{i,1})=(c^{j}_{i,0},c^{j}_{i,1})$ for some, equivalently all, $j\geq i$. Then $(a_{i},c^{\omega}_{i,0},c^{\omega}_{i,1})_{i<\omega}$ satisfies conditions (1)—(3) so, after extracting an indiscernible sequence, we conclude. ∎

Remark 4.2*.*

If $\varphi(x;y)$ witnesses SOP1 in $T$, it is clear from the definition that $\varphi$ will witness SOP1 in any expansion $T^{\prime}$ of $T$ and hence we may apply the above lemma to find $(a_{i},c_{i,0},c_{i,1})_{i<\omega}$ which are morever $L^{\text{Sk}}$-indiscernible and satisfy $c_{i,0}\equiv^{L^{Sk}}_{a_{<i},c_{<i,0}c_{<i,1}}c_{i,1}$ for all $i$ in $\mathbb{M}^{Sk}$, where the $L^{\text{Sk}}$-structure $\mathbb{M}^{\text{Sk}}$ is a monster model of an expansion of $T$ with Skolem functions. See, e.g., [KR20, Remark 2.5].

Proposition 4.3**.**

Suppose $T$ has SOP1. Then there are models $M\prec N\models T$ and tuples $a$ and $c$ such that a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{M}N, a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{N}c and a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}^{K}_{M}Nc.

Proof.

Fix a Skolemization $T^{\text{Sk}}$ of $T$ in the language $L^{\text{Sk}}$ and work in a monster model $\mathbb{M}^{\text{Sk}}\models T^{\text{Sk}}$. We will write $\equiv^{L^{\mathrm{Sk}}}$ to denote equality of types in the language $L^{\mathrm{Sk}}$ and $\equiv$ to denote equality of types in the language $L$. By Lemma 4.1 and compactness, we can find an $L$-formula $\varphi(x;y)$ and an $L^{\text{Sk}}$-indiscernible sequence $(a_{i},c_{i,0},c_{i,1})$ such that

1. (1)

For all $i\in\mathbb{Q}$, $a_{i}\models\{\varphi(x;c_{j,0}):j\leq i\}$. 2. (2)

$\{\varphi(x;c_{i,1}):i\in\omega\}$ is $2$-inconsistent. 3. (3)

For all $i\in\mathbb{Q}$, $c_{i,0}\equiv^{L^{\text{Sk}}}_{a_{<i},c_{<i,0}c_{<i,1}}c_{i,1}$.

Define $M=\text{Sk}(a_{<0}c_{<0,0}c_{<0,1})$ and $N=\text{Sk}(a_{<0},c_{<0,0},c_{<0,1},a_{>1})$. Note that we have $M\prec N$. In the claims below, independence is understood to mean independence with respect to the $L$-theory $T$.

Claim 1: a_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{M}N.

Proof of claim: Fix a formula $\psi(x;n)\in\text{tp}(a_{1}/N)$. We can write the tuple $n=t(a,c)$ where $t$ is a tuple of Skolem terms, $a$ is a finite tuple from $a_{<0}a_{>1}$ and $c$ is a finite tuple from $c_{<0,0}c_{<0,1}$. As $a$ and $c$ are finite, there is some rational $\epsilon<0$ such that $a$ and $c$ come from $a_{<\epsilon}a_{>1}$ and $c_{<\epsilon,0}c_{<\epsilon,1}$ respectively. By indiscernibility, $\psi(x;n)$ is realized also by any $a_{\delta}$ with $\epsilon<\delta<0$, which is in $M$.∎

Claim 2: a_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{N}c_{0,0}.

Proof of claim: This has a similar proof to Claim 1. Given any $\psi(x;n,c_{0,0})\in\text{tp}(a_{1}/Nc_{0,0})$, as before, we can write the tuple $n=t(a,c)$ where $t$ is a tuple of Skolem terms, $a$ is a finite tuple from $a_{<0}a_{>1}$ and $c$ is a finite tuple from $c_{<0,0}c_{<0,1}$. Because these tuples are finite, there is a rational $\epsilon>1$ such that $a$ comes from $a_{<0}a_{>\epsilon}$. Then by indiscernibility, $\psi(x;n,c_{0,0})$ is satisfied by any $a_{\delta}$ with $1<\delta<\epsilon$, all of which are in $N$. ∎

Claim 3: a_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}^{K}_{M}Nc_{0,0}.

Proof of claim: We will show even a_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}^{K}_{M}c_{0,0}. Let $\mathcal{D}$ be an ultrafilter on $M$ containing $\{c_{i,1}:i\in(\epsilon,0)\}$ for every $\epsilon<0$. By $L^{\mathrm{Sk}}$-indiscernibility, we have $c_{0,1}\models\text{Av}(\mathcal{D},M)$. Then there is a sequence $(b_{i})_{i<\omega}\models\text{Av}(\mathcal{D},\mathbb{M}^{\text{Sk}})^{\otimes\omega}|_{M}$ with $b_{0}=c_{0,1}$. By (2) and the choice of $\mathcal{D}$, we know $\{\varphi(x;b_{i}):i<\omega\}$ is $2$-inconsistent so $\varphi(x;c_{0,1})$ Kim-divides over $M$. Moreover, $c_{0,0}\equiv^{L^{\text{Sk}}}_{a_{<0}c_{<0,0}c_{<0,1}}c_{0,1}$ so, in particular, $c_{0,0}\equiv_{M}c_{0,1}$ from which it follows also that $\varphi(x;c_{0,0})$ Kim-divides over $M$. By (1), we have $\models\varphi(a_{1},c_{0,0})$ so a_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}^{K}_{M}c_{0,0}. ∎

The claims taken together show a_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{n}_{M}N, a_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{N}c_{0,0}, and a_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}^{K}_{M}c_{0,0}, which completes the proof. ∎

Corollary 4.4**.**

The following are equivalent:

1. (1)

$T$* is NSOP1.* 2. (2)

\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}* satisfies the following weak form of transitivity: if $M\prec N\models T$, a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{M}N and a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{N}b, then a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Nb.* 3. (3)

\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}* satisfies transitivity: if $M\prec N\models T$, a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N and a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}b, then a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}Nb.*

Proof.

Theorem 3.4 establishes (1)$\implies$(3), Proposition 4.3 shows (2)$\implies$(1), and (3)$\implies$(2) is immediate from the fact that \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u} implies \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}. ∎

5. \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequences are witnesses

In this section, we characterize NSOP1 by the property that \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequences are witnesses to Kim-dividing. The non-structure direction of this characterization was already observed in [KR20, Theorem 3.16]: if $T$ has SOP1 then \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequences will not always witness Kim-dividing. The more interesting direction goes the other way, showing that in the NSOP1 context, \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequences are witnesses. This is a significant technical development in the study of NSOP1 theories, as it, for example, obviates the need in many cases to construct tree Morley sequences. We give some applications below.

Theorem 5.1**.**

Suppose that $\varphi\left(x,a\right)$ Kim-divides over $M$. Suppose that $\langle a_{i}:i<\omega\rangle$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$, starting with $a$. Then $\{\varphi\left(x,a_{i}\right):i<\omega\}$ is inconsistent.

Proof.

Suppose not. Let $\kappa=\left|M\right|+|T|$ and extend the sequence to have length $\kappa^{+}$. It suffices to find an increasing continuous sequence of models $\langle N_{i}:i<\kappa^{+}\rangle$ such that $N_{i}$ contains $a_{<i}$, $\left|N_{i}\right|\leq\kappa$, $N_{0}=M$ and such that a_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}^{K}N_{i}. To see this, suppose that $c\models\{\varphi\left(x,a_{i}\right):i<\kappa^{+}\}$. Then by local character, Remark 2.8, for some $i<\kappa^{+}$, c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{N_{i}}^{K}N_{\kappa^{+}} where $N_{\kappa^{+}}=\bigcup_{i<\kappa^{+}}N_{i}$, as $\{N_{i}:\kappa\leq i<\kappa^{+}\}$ is a club subset of $[N_{\kappa^{+}}]^{\kappa}$. Hence c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{N_{i}}^{K}a_{i}. However, a_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}^{K}N_{i} and hence by transitivity and symmetry, N_{i}c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}^{K}a_{i} contradicting our assumption that $\varphi\left(x,a\right)$ Kim-divides over $M$ and hence also $\varphi(x;a_{i})$, by invariance.

Claim: There is a partial type $\Gamma(\overline{x})$ over $a_{<\kappa^{+}}M$ such that:

1. (1)

We have $\overline{x}=\langle\overline{x}_{\alpha}:\alpha<\kappa^{+}\rangle$ is an increasing continuous sequence of tuples of variables such that $\left|\overline{x}_{\alpha}\right|=\kappa$, and such that $\overline{x}_{\alpha+1}$ contains $\kappa$ new variables not in $\overline{x}_{\alpha}$ for all $\alpha<\kappa^{+}$. 2. (2)

$\Gamma\left(\overline{x}\right)$ asserts that $\overline{x}_{\alpha}$ enumerates a model containing $Ma_{<\alpha}$ for all $\alpha<\kappa^{+}$.

Proof of claim: We define $\Gamma(\overline{x})$ as a continuous increasing union of partial types $\Gamma_{\alpha}(\overline{x}_{\alpha})$ for $\alpha<\kappa^{+}$. Suppose we are given $\Gamma_{\delta}(\overline{x}_{\delta})$ for $\delta<\alpha$.

If $\alpha=\beta+1$, then we define $\overline{x}_{\alpha,0}=\overline{x}_{\beta}$ and then, given $\overline{x}_{\alpha,i}$, we define $\Lambda_{i}$ to be the set of all partitioned formulas $\varphi(y;\overline{x})$ where the parameters of $\varphi$ come from $a_{<\alpha}M$ and the parameter variables $\overline{x}$ of $\varphi$ are among $\overline{x}_{\alpha,i}$. Now define $\overline{x}_{\alpha,i+1}=\overline{x}_{\alpha,i}$ together with a new variable $x_{\lambda}$ for each $\lambda\in\Lambda_{i}$. Finally $\overline{x}_{\alpha}=\bigcup_{i<\omega}\overline{x}_{\alpha,i}$. Let $\Gamma_{\alpha,0}=\Gamma_{\beta}$ and, given $\Gamma_{\alpha,i}$, we define $\Gamma_{\alpha,i+1}$ by

[TABLE]

Then $\Gamma_{\alpha}(\overline{x}_{\alpha})=\bigcup_{i<\omega}\Gamma_{\alpha,i}(\overline{x}_{\alpha,i})$. Note that because $\mathbb{M}\models(\exists y)[y=c]$ for each $c\in a_{<\alpha}M$, any realization of $\Gamma_{\alpha}(\overline{x}_{\alpha})$ will contain $a_{<\alpha}M$ and will be a model by the Tarski-Vaught test.

To complete the induction, we note that if $\alpha$ is a limit and we are given $\Gamma_{\delta}$ for all $\delta<\alpha$, then we can set $\overline{x}_{\alpha}=\bigcup_{\delta<\alpha}\overline{x}_{\delta}$ and $\Gamma_{\alpha}(\overline{x}_{\alpha})=\bigcup_{\delta<\alpha}\Gamma_{\delta}(\overline{x}_{\delta})$, which has the desired property as the union of an elementary chain is a model. ∎

Lastly, we define $\Delta(\overline{x})$ as follows:

[TABLE]

where we write $\varphi(\overline{x}_{\alpha};a_{\alpha})$ to denote a formula whose variables are a finite subtuble of $\overline{x}_{\alpha}$. To conclude, it is enough, by symmetry, to show that $\Delta\left(\overline{x}\right)$ is consistent. By compactness, it is enough to prove this when replace $\kappa^{+}$ by a natural number $n<\omega$, so we prove it by finding such a sequence by induction on $n$. Suppose we found such an increasing sequence of models $N_{i}$ for $i<n$. Let $N_{n}$ be a model containing $MN_{n-1}a_{<n}$ of size $\kappa$. Since a_{n}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}a_{<n}, we may assume by extension that a_{n}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}N_{n}, preserving all the previous types, so we are done. ∎

Corollary 5.2**.**

Suppose $T$ is NSOP1 and $M\models T$. If $\langle a_{i}:i<\omega\rangle$ is a Kim-Morley sequence over $M$ starting with $a_{0}=a$, then $\varphi\left(x,a\right)$ Kim-divides over $M$ iff $\{\varphi\left(x,a_{i}\right):i<\omega\}$ is $k$-inconsistent for some $k<\omega$.

Proof.

One direction is Fact 2.9(2). The other is Theorem 5.1, since, by compactness and indiscernibility, if $\{\varphi(x;a_{i}):i<\omega\}$ is inconsistent, it is $k$-inconsistent for some $k<\omega$. ∎

Remark 5.3*.*

In fact, in Corollary 5.2 we only need to assume $\langle a_{i}:i<\omega\rangle$ satisfies a_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}a_{<i} and $a_{i}\models\text{tp}(a_{i}/M)$ for all $i<\omega$ (i.e. it is not necessary to assume that this sequence is $M$-indiscernible). If $\varphi(x;a)$ does not Kim-divide over $M$, then $\{\varphi(x;a_{i}):i<\omega\}$ is consistent by the independence theorem over $M$. Conversely, if $\{\varphi(x;a_{i}):i<\omega\}$ is not $k$-inconsistent for any $k<\omega$, then the partial type $\Gamma(y_{i}:i<\omega)$ containing, for all $i<\omega$,

• •

$y_{i}\models\text{tp}(a/M)$,

• •

$\{\psi(y_{<i};y_{i}):\psi(y_{<i};a)\in L(Ma)\text{ Kim-forks over }M\}$

• •

$(\exists x)\bigwedge_{j<i}\varphi(x;y_{i})$

together with a schema asserting $\langle y_{i}:i<\omega\rangle$ is $M$-indiscernible is finitely satisfiable in $\langle a_{i}:i<\omega\rangle$ by compactness, Ramsey, and symmetry. A realization contradicts Corollary 5.2.

Corollary 5.4**.**

Suppose $T$ is NSOP1, $M\models T$, and $I=\langle a_{i}:i<\omega\rangle$ is an $M$-indiscernible sequence. The $I$ is a witness for Kim-dividing over $M$ if and only if $I$ is a \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$.

Proof.

Note that if $I$ is a witness for Kim-dividing over $M$, then a_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}a_{<i} for all $i<\omega$ by symmetry: if $\varphi(x;a_{i})\in\text{tp}(a_{<i}/Ma_{i})$, then, by $M$-indiscernibility, $a_{<i}\models\{\varphi(x;a_{j}):j\geq i\}$ so $\varphi(x;a_{i})$ does not Kim-divide over $M$, hence a_{<i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}a_{i}. This shows that witnesses for Kim-dividing over $M$ are \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley over $M$. The other direction is Theorem 5.1. ∎

Corollary 5.5**.**

Suppose $T$ is NSOP1 and $M\models T$. A sequence $I$ over $M$ is tree Morley over $M$ if and only if $I$ is a total \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$.

Proof.

By Fact 2.22(1), if $I$ is tree Morley over $M$ then $I$ is a total Morley sequence over $M$. For the other direction, suppose $I=\langle a_{i}:i<\omega\rangle$ is a total \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$ and we will show it is tree Morley over $M$. By Fact 2.19, it suffices to show $I$ is a strong witness to Kim-dividing over $M$. Because a_{>i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}a_{\leq i} for all $i<\omega$, if $1\leq n<\omega$ we know $\langle(a_{n\cdot i},a_{n\cdot i+1},\ldots,a_{n\cdot i+(n-1)}\rangle$ satisfies

[TABLE]

for all $i<\omega$, or, in other words, $\langle(a_{n\cdot i},a_{n\cdot i+1},\ldots,a_{n\cdot i+(n-1)}):i<\omega\rangle$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$, hence a witness to Kim-dividing over $M$ by Theorem 5.1. It follows that $I$ is a strong witness to Kim-dividing, so $I$ is tree Morley over $M$. ∎

6. Applications

6.1. Lifting lemmas

The first application of the transitivity and witnessing theorems will be two ‘lifting lemmas’ that concern \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley and tree Morley sequences over two bases simultaneously. In Lemma 3.1, we showed that if $M\prec N$ and a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N, then it is possible to construct an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$ beginning with $a$ which is also a tree Morley sequence over $N$. Later, we showed under the same hypotheses in Proposition 3.3, that we can construct a tree Morley sequence over $M$ starting with $a$ which is also an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $N$. These raise two natural questions: first, is it possible, under these hypotheses, to construct sequences that are tree-Morley over both bases simultaneously? And if so, are such sequences somehow special? We show that the answer to the first question is yes, and, moreover, address the second by showing that every \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence (tree-Morley sequence) over $M$ beginning with $a$ is conjugate over $Ma$ to a sequence that is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley (tree Morley) over $N$.

Definition 6.1**.**

We say that $(a_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-spread out over $M$ if for all $\eta\in\mathcal{T}_{\alpha}$ with $\text{dom}(\eta)=[\beta+1,\alpha)$ for some $\beta<\alpha$, the sequence $(a_{\unrhd\eta\frown\langle i\rangle})_{i<\omega}$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$.

Lemma 6.2**.**

Suppose $(a_{\eta})_{\eta\in\mathcal{T}_{\kappa}}$ is a tree of tuples, \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-spread out and $s$-indiscernible over $M$. If $\kappa$ is sufficiently large, then there is a tree $(b_{\eta})_{\eta\in\mathcal{T}_{\omega}}$, $s$-indiscernible and \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-spread out over $M$, such that:

1. (1)

For all $w\in[\omega]^{<\omega}$,

[TABLE]

for some $v\in[\kappa\setminus\text{lim}(\kappa)]^{<\omega}$. 2. (2)

For all $w,v\in[\omega]^{<\omega}$ with $|w|=|v|$,

[TABLE]

Proof.

The proof of [KR20, Lemma 5.10] (Fact 2.17(2)) shows that there is $(b_{\eta})_{\eta\in\mathcal{T}_{\omega}}$ satisfying (1) and (2). As $(a_{\eta})_{\eta\in\mathcal{T}_{\kappa}}$ is $s$-indiscernible and \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-spread out over $M$, (1) implies that $(b_{\eta})_{\eta\in\mathcal{T}_{\omega}}$ is $s$-indiscernible and \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-spread out over $M$ as well. See the proof of [KR20, Lemma 5.10] (Fact 2.17(2)) for more details. ∎

Lemma 6.3**.**

Suppose $M$ is a model and $(a_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ is a tree which is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-spread out and $s$-indiscernible over $M$ and for all $w,v\in[\alpha\setminus\text{lim}(\alpha)]^{<\omega}$ with $|w|=|v|$,

[TABLE]

then $(a_{\zeta_{\beta}})_{\beta\in\alpha\setminus\text{lim}(\alpha)}$ is a tree Morley sequence over $M$.

Proof.

The condition that for all $w,v\in[\alpha\setminus\text{lim}(\alpha)]^{<\omega}$ with $|w|=|v|$,

[TABLE]

implies that $(a_{\zeta_{\beta}})_{\beta\in\alpha\setminus\text{lim}(\alpha)}$ is an $M$-indiscernible sequence. By Corollary 5.5, it suffices to show that $(a_{\zeta_{\beta}})_{\beta\in\alpha\setminus\text{lim}(\alpha)}$ is a total \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$. Fix any non-limit $\beta<\alpha$. We know that $a_{\zeta_{\leq\beta}}$ is a subtuple of $a_{\unrhd\zeta_{\beta}}=a_{\vartriangleright\zeta_{\beta+1\frown 0}}$ and $\langle a_{\vartriangleright\zeta_{\beta+1\frown\langle i\rangle}}:i<\omega\rangle$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$ which is $Ma_{\zeta_{>\beta}}$-indiscernible so a_{\zeta_{>\beta}}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}a_{\zeta_{\leq\beta}} by Theorem 5.1. ∎

Proposition 6.4**.**

Suppose $T$ is NSOP1, $M\prec N\models T$, and $I=\langle b_{i}:i<\omega\rangle$ is a tree Morley sequence over $M$. If b_{0}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N, then there is $I^{\prime}\equiv_{Mb_{0}}I$ such that $I^{\prime}$ is a tree Morley sequence over $N$.

Proof.

By compactness, we may stretch the sequence so that $I=\langle b_{i}:i\in\kappa\setminus\text{lim}(\kappa)\rangle$ for some cardinal $\kappa$ large relative to $|N|$. By the chain condition, Lemma 2.10, we may also assume $I$ is $N$-indiscernible and I\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N after moving by an automorphism over $Mb_{0}$. By induction on $\alpha\leq\kappa$, we will construct trees $(b^{\alpha}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ and sequences $I_{\alpha}=\langle b_{\alpha,i}:i\in\kappa\setminus\text{lim}(\kappa)\rangle$ satisfying the following conditions for all $\alpha$:

1. (1)

For all non-limit $i\leq\alpha$, $b^{\alpha}_{\zeta_{i}}=b_{\alpha,i}$. 2. (2)

$(b^{\alpha}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-spread out over $N$ and $s$-indiscernible over $NI_{\alpha,>\alpha}$. 3. (3)

If $\beta<\alpha$, $I_{\alpha}\equiv_{M}I_{\beta}$ and $I_{0}=I$. 4. (4)

$I_{\alpha,>\alpha}$ is $M(b^{\alpha}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$-indiscernible. 5. (5)

If $\alpha<\beta$, then $b^{\alpha}_{\eta}=b^{\beta}_{\iota_{\alpha\beta}(\eta)}$ for $\eta\in\mathcal{T}_{\alpha}$. 6. (6)

I_{\alpha}(b^{\alpha}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N. 7. (7)

$b^{\alpha}_{\eta}\equiv_{N}b_{\alpha,i}\equiv_{N}b_{0}$ for all $\eta\in\mathcal{T}_{\alpha}$, $i\in\kappa\setminus\text{lim}(\kappa)$.

For the base case, we define $b^{0}_{\emptyset}=b_{0}$ and $I_{0}=I$, which satisfies all the demands. Next, suppose we are given $(b^{\beta}_{\eta})_{\eta\in\mathcal{T}_{\beta}}$ for all $\beta\leq\alpha$ and we will construct $(b^{\alpha+1}_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}$. By (6) and Lemma 3.3, we may obtain a sequence $J=\langle(b^{\alpha}_{\eta,i})_{\eta\in\mathcal{T}_{\alpha}}:i<\omega\rangle$ with $(b^{\alpha}_{\eta,0})_{\eta\in\mathcal{T}_{\alpha}}=(b^{\alpha}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ which is tree-Morley over $M$ and \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley over $N$. As $J$ is a tree Morley sequence over $M$ which is $N$-indiscernible, we have:

[TABLE]

Likewise, as $I_{\alpha,>\alpha}$ is a tree Morley sequence over $M$ by (3) which is $M(b^{\alpha}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$-indiscernible by (4), we have I_{\alpha,>\alpha}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}(b^{\alpha}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}. By the chain condition (Lemma 2.10), there is $I^{\prime}_{\alpha,>\alpha}\equiv_{M(b^{\alpha}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}}I_{\alpha,>\alpha}$ so that $J$ is $MI^{\prime}_{\alpha,>\alpha}$-indiscernible and also:

[TABLE]

Choose $N^{\prime}$ so that $NI_{\alpha,>\alpha}\equiv_{M}N^{\prime}I^{\prime}_{\alpha,>\alpha}$. By (6) and invariance, we have

[TABLE]

By (a), (b), and (c), we may apply the independence theorem to find a model $N^{\prime\prime}$ with $N^{\prime\prime}\equiv_{MJ}N$, $N^{\prime\prime}\equiv_{MI^{\prime}_{\alpha,>\alpha}}N^{\prime}$, and N^{\prime\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}I^{\prime}_{\alpha,>\alpha}J. Now choose $I^{\prime\prime}_{\alpha,>\alpha}=\langle b^{\prime\prime}_{\alpha,i}:i\in\kappa\setminus(\text{lim}(\kappa)\cup\alpha)\rangle$ so that $NI^{\prime\prime}_{\alpha,>\alpha}\equiv_{MJ}N^{\prime\prime}I^{\prime}_{\alpha,>\alpha}$.

Define a tree $(c_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}$ by setting $c_{\emptyset}=b^{\prime\prime}_{\alpha,\alpha+1}$ and $c_{\langle i\rangle\frown\eta}=b^{\alpha}_{\eta,i}$ for all $\eta\in\mathcal{T}_{\alpha}$ and $i<\omega$. With this definition, we have N\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}(c_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}I^{\prime\prime}_{\alpha,>\alpha+1}. Let $(c^{\prime}_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}$ be a tree which is $s$-indiscernible over $NI^{\prime\prime}_{\alpha,>\alpha+1}$ locally based on $(c_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$. By symmetry and finite character, we have N\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}(c^{\prime}_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}I^{\prime\prime}_{\alpha,>\alpha+1}. Finally, let $I^{\prime\prime\prime}_{\alpha,>\alpha+1}=\langle b^{\prime\prime\prime}_{\alpha,i}:i\in\kappa\setminus(\text{lim}(\kappa)\cup(\alpha+1))\rangle$ be an $N(c^{\prime}_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}$-indiscernible sequence locally based on $I^{\prime\prime}_{\alpha,>\alpha+1}$. By symmetry, we have N\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}I^{\prime\prime\prime}_{\alpha,>\alpha}(c^{\prime}_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}. Note that, by (2) and the construction, $(c^{\prime}_{\eta})_{\eta\in\mathcal{T}_{\alpha+1}}$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-spread out over $N$ and $s$-indiscernible over $NI^{\prime\prime\prime}_{\alpha,>\alpha+1}$. Moreover, by (2) and the construction, there is an automorphism $\sigma\in\text{Aut}(\mathbb{M}/N)$ such that $\sigma(c^{\prime}_{\langle 0\rangle\frown\eta})=b^{\alpha}_{\eta}$ for all $\eta\in\mathcal{T}_{\alpha}$ so we will define $b^{\alpha+1}_{\eta}=\sigma(c^{\prime}_{\eta})$ for all $\eta\in\mathcal{T}_{\alpha+1}$. Likewise, we define $I_{\alpha+1}=\langle b_{\alpha+1,i}:i\in\kappa\setminus\text{lim}(\kappa)\rangle$ by $b_{\alpha+1,i}=b^{\alpha+1}_{\zeta_{i}}$ for non-limit $i\leq\alpha+1$ and $b_{\alpha+1,i}=\sigma(b^{\prime\prime\prime}_{\alpha,i})$ for non-limit $i>\alpha+1$. It is immediate that this construction satisfies (6) and (7) by induction and the construction of $N^{\prime\prime}$. To check (3), note that, by induction, using (1),(2), and (3), for any function $\eta:\alpha\to\omega$, we have $(b^{\alpha}_{\eta|_{[\beta,\alpha)}})_{\beta\in\alpha\setminus\text{lim}(\alpha)}I_{\alpha,>\alpha}\equiv_{M}I$, and therefore, for any $i<\omega$, we have

[TABLE]

By the definition of $(c_{\eta})_{\eta\in T_{\alpha+1}}$ and $s$-indiscernibility over $M$, it follows that, for any function $\eta^{\prime}:(\alpha+1)\to\omega$,

[TABLE]

from which (3) follows. The remaining constraints are easily seen to be satisfied by the construction.

Now for $\delta$ limit, if we are given $(b^{\alpha}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$ for $\alpha<\delta$, we may define $b^{\delta}_{\iota_{\alpha\delta}(\eta)}=b^{\alpha}_{\eta}$ for all $\alpha<\delta$ and $\eta\in\mathcal{T}_{\alpha}$. We define $I_{\delta}$ as follows: $I_{\delta,<\delta}$ will be defined by $b_{\delta,i}=b_{i,i}$ for all non-limit $i<\delta$. By (1),(3), and induction, we have $I_{\delta,<\delta}\equiv_{M}I_{<\delta}$. Choose $J$ so that $I_{\delta,<\delta}J\equiv_{M}I_{<\delta}I_{>\delta}$. Write $\overline{x}$ for $\langle x_{i}:i\in\kappa\setminus(\text{lim}(\kappa)\cup\delta)\rangle$ and $\varphi(\overline{x};c,n)$ to denote any formula where the variables are a finite subtuple of $\overline{x}$. By (6), induction, and compactness, the partial type, which contains $\text{tp}_{\overline{x}}(J/MI_{\delta,<\delta})$ and $\{\neg\varphi(\overline{x},c;n):c\in(b^{\delta}_{\eta})_{\eta\in\mathcal{T}_{\delta}},n\in N,\varphi(x,y;n)\text{ Kim-divides over }M\}$, and naturally expresses that both $(b^{\delta}_{\eta})_{\eta\in\mathcal{T}_{\delta}}$ is $s$-indiscernible over $N\overline{x}$ and $\overline{x}$ is $N(b^{\delta}_{\eta})_{\eta\in\mathcal{T}_{\alpha}}$-indiscernible, is consistent. Let $I_{\delta,>\delta}$ is a realization of this type, completing the definition of $I_{\delta}$. It is easy to check that these are well-defined and satisfy all of the requirements by induction and the finite character of Kim-independence.

This completes the recursion and yields $(b^{\kappa}_{\eta})_{\eta\in\mathcal{T}_{\kappa}}$ likewise defined by $b^{\kappa}_{\iota_{\alpha\kappa}(\eta)}=b^{\alpha}_{\eta}$ for all $\alpha<\kappa$ and $\eta\in\mathcal{T}_{\alpha}$. Apply Lemma 6.2 to obtain a tree $(c_{\eta})_{\eta\in\mathcal{T}_{\omega}}$ so that for all $w\in[\omega]^{<\omega}$, there is $v\in[\kappa\setminus\text{lim}(\kappa)]^{<\omega}$ such that

[TABLE]

and, moreover, for all $w,v\in[\omega]^{<\omega}$ with $|w|=|v|$,

[TABLE]

By an automorphism, we can assume $c_{\zeta_{0}}=b_{0}$, hence, setting $I^{\prime}=\langle c_{\zeta_{i}}:i<\omega\rangle$, we have $I^{\prime}\equiv_{Mb_{0}}I$. Moreover, by Lemma 6.3, $I^{\prime}$ is a tree Morley sequence over $N$, completing the proof. ∎

The second lifting lemma, is an analogue of Proposition 6.4 for \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequences.

Proposition 6.5**.**

Suppose $T$ is NSOP1, $M\prec N\models T$, and $I=\langle b_{i}:i<\omega\rangle$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$. If b_{0}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N, then there is $I^{\prime}\equiv_{Mb_{0}}I$ satisfying the following conditions:

1. (1)

I^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N** 2. (2)

$I^{\prime}$* is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $N$.*

Proof.

For $i<\omega$, let $q_{i}(x_{j}:j\leq i)=\text{tp}(b_{\leq i}/M)$ and let $p(x;N)=\text{tp}(b_{0}/N)$. For a natural number $K$, define $\Gamma_{K}$ to be the partial type defined as the union of the following:

1. (a)

$q_{K}(x_{i}:i\leq K)$. 2. (b)

$\bigcup_{i\leq K}p(x_{i};N)$. 3. (c)

$\{\neg\varphi(x_{\leq K};c):\varphi(x_{\leq K},y)\in L(M),c\in N,\varphi(x_{\leq K};c)\text{ Kim-divides over }M\}$. 4. (d)

$\{\neg\varphi(x_{<i};x_{i}):i\leq K,\varphi(x_{<i};x_{i})\in L(N),\varphi(x_{<i};b_{0})\text{ Kim-divides over }N\}$.

By Ramsey and compactness, it is enough to show the consistency of $\Gamma=\bigcup_{K<\omega}\Gamma_{K}$.

As b_{0}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N, $\Gamma_{0}$ is consistent. Suppose $\Gamma_{K}$ is consistent and we will that show $\Gamma_{K+1}$ is consistent. Let $\Delta(x_{0},\ldots,x_{K+1})\subseteq\Gamma_{K+1}$ be the partial type defined as the union of the following:

1. (1)

$q_{K+1}(x_{i}:i\leq K+1)$. 2. (2)

$\bigcup_{i\leq K+1}p(x_{i};N)$. 3. (3)

$\{\neg\varphi(x_{\leq K+1};c):\varphi(x_{\leq K+1},y)\in L(M),c\in N,\varphi(x_{\leq K+1};c)\text{ Kim-divides over }M\}$. 4. (4)

$\{\neg\varphi(x_{<i};x_{i}):i<K,\varphi(x_{<i};x_{i})\in L(N),\varphi(x_{<i};b_{0})\text{ Kim-divides over }N\}$.

Note that $\Delta$ is identical to $\Gamma_{K+1}$ except that in the final set of formulas, $i$ is taken to be less than $K$ rather than $K+1$.

Claim 1: $\Delta$ is consistent.

Proof of claim: Let $(b^{\prime}_{0},\ldots,b^{\prime}_{K})\models\Gamma_{K}$ and choose $b^{\prime}_{K+1}$ so that $b^{\prime}_{\leq K+1}\equiv_{M}b_{\leq K+1}$. Next, choose a model $N^{\prime}$ so that $b^{\prime}_{K+1}N^{\prime}\equiv_{M}b_{0}N$. Now by definition of $\Gamma_{K}$ and symmetry, we have N\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b^{\prime}_{\leq K} and our assumption that b_{0}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N implies N^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b^{\prime}_{K+1} by symmetry and invariance. Moreover, because $I$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence, we likewise have b^{\prime}_{K+1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b^{\prime}_{\leq K}. Therefore, we may apply the independence theorem to find $N^{\prime\prime}\models\text{tp}(N/Mb^{\prime}_{\leq K})\cup\text{tp}(N^{\prime}/Mb^{\prime}_{K+1})$ such that N^{\prime\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b^{\prime}_{\leq K+1}. There is an automorphism $\sigma\in\text{Aut}(\mathbb{M}/Mb^{\prime}_{\leq K})$ with $\sigma(N^{\prime\prime})=N$. Let $b^{\prime\prime}_{K+1}=\sigma(b^{\prime}_{K+1})$. Then $(b^{\prime}_{0},\ldots,b^{\prime}_{K},b^{\prime\prime}_{K+1})\models\Delta$.∎

Claim 2: Suppose $J=\langle c_{K+1,i}:i<\omega\rangle$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$ with $b_{0}=c_{K+1,0}$. If $J$ is $N$-indiscernible and J\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N, then $\bigcup_{i<\omega}\Delta(x_{0},\ldots,x_{K},c_{K+1,i})$ is consistent.

Proof of claim: Choose $(c_{0},\ldots,c_{K})$ so that $(c_{0},\ldots,c_{K},c_{K+1,0})\models\Delta$. Then c_{K+1,0}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}c_{\leq K} so there is $J^{\prime}\equiv_{Mc_{K+1,0}}J$ such that $J^{\prime}$ is $Mc_{\leq K}$-indiscernible and J^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}c_{\leq K}, by the chain condition for \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequences Lemma 2.10. Moreover, by definition of $\Delta$, we have c_{\leq K}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N. By assumption, J\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N and, since $J\equiv_{M}J^{\prime}$, we may, therefore, apply the strengthened independence theorem, Fact 2.9(3), to find $J_{*}$ that simultaneously realizes $\text{tp}(J^{\prime}/Mc_{\leq K})$, to satisfy condition $(1)$ in the definition of $\Delta$, and $\text{tp}(J/N)$, to satisfy condition (2), and, moreover, such that N\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}J_{*}c_{\leq K}, to satisfy (3). Choose $c^{\prime}_{\leq K}$ so that $c^{\prime}_{\leq K}J\equiv_{N}c_{\leq K}J_{*}$. Then, by definition of $\Delta$, $c^{\prime}_{\leq K}\models\bigcup_{i<\omega}\Delta(x_{0},\ldots,x_{K},c_{K+1,i})$.∎

To conclude, we use Proposition 6.4 to select $J=\langle b^{\prime}_{K+1,i}:i<\omega\rangle$ which is simultaneously a tree Morley sequence over $M$ and a tree Morley sequence over $N$ with J\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N. In particular, $J$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$. Hence, by Claim 2,

[TABLE]

is consistent, so we may realize it with $(b^{\prime}_{0},\ldots,b^{\prime}_{K})$. By compactness and Ramsey, we may additionally assume that $\langle b_{K+1,i}:i<\omega\rangle$ is $Nb^{\prime}_{\leq K}$-indiscernible. Put $b^{\prime}_{K+1}=b^{\prime}_{K+1,0}$. It follows, by Kim’s Lemma for tree Morley sequences (Fact 2.19(1)), that b^{\prime}_{K+1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}b^{\prime}_{\leq K} and, therefore by definition of $\Delta$, b^{\prime}_{<i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{N}b^{\prime}_{i} for all $i\leq K+1$. Additionally, by definition of $\Delta$, we have $b^{\prime}_{\leq K+1}\models q_{K+1}$, $b^{\prime}_{i}\models p(x;N)$ for all $i\leq K+1$, and b^{\prime}_{\leq K+1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}N. This shows $b^{\prime}_{\leq K+1}\models\Gamma_{K+1}$. ∎

6.2. Doubly local character

In [KRS19, Lemma 3.7], the following variant of local character was established: if $\langle M_{i}:i<\alpha\rangle$ is an increasing sequence of elementary submodels of $N$ and $p\in S(N)$ does not Kim-divide over $M_{i}$ for all $i<\alpha$, then $p$ does not Kim-divide over $M_{\alpha}$. The proof there uses the fact that $p$ is a complete type in an essential way, which left open whether or not a local version of this form of local character (hence the name doubly local character) might also hold, where the type $p$ is replaced by a formula over $N$. We prove this in Proposition 6.10, answering [KRS19, Question 3.17] .

Definition 6.6**.**

Suppose $\alpha$ is an ordinal and $\mathcal{U}$ is an ultrafilter on $\alpha$. Given a sequence of sequences $\langle\overline{b}_{i}:i<\omega\rangle$, where $\overline{b}_{i}=\langle b_{i,j}:j<\omega\rangle$ for all $i<\alpha$, we say that $\overline{a}$ is a $\mathcal{U}$-average of $\langle\overline{b}_{i}:i<\alpha\rangle$ over $A$ if, for all $n<\omega$ and $\varphi(x_{0},\ldots,x_{n-1})\in L(A)$, we have

[TABLE]

It is an easy exercise to show that $\mathcal{U}$-averages exist for any sequence of sequences and parameter sets $A$.

Lemma 6.7**.**

Suppose we are given:

1. (1)

An increasing continuous elementary chain $\langle M_{i}:i\leq\alpha\rangle$ of models of $T$. 2. (2)

For every $i<\alpha$, $\bar{b}_{i}=\langle b_{i,j}:j<\omega\rangle$ is an indiscernible heir sequence over $M_{i}$. 3. (3)

For all $i\leq j$, $b_{i,0}\equiv_{M_{i}}b_{j,0}$.

Then for any ultrafilter $\mathcal{U}$ on $\alpha$ concentrating on end segments of $\alpha$, if $\bar{a}=\langle a_{j}:j<\omega\rangle$ realizes the $\mathcal{U}$-average of $\langle\bar{b}_{i}:i<\alpha\rangle$ over $M_{\alpha}$, then $\langle a_{j}:j<\omega\rangle$ is an heir sequence over $M_{\alpha}$ such that $a_{0}\equiv_{M_{i}}b_{i,0}$ for all $i<\alpha$.

Proof.

The fact that $\bar{a}$ is an indiscernible sequence over $M_{\alpha}$ and $a_{0}\equiv_{M_{i}}b_{i,0}$ is clear by construction. We are left with showing that $\bar{a}$ is an heir sequence over $M_{\alpha}$. Suppose that $\psi\left(a_{j},a_{<j},m\right)$ where $m\in M_{\alpha}$ and $\psi\left(y,z,w\right)$ is an $L$-formula. Then for some $i<\alpha$ such that $m\in M_{i}$, $\psi\left(b_{i,j},b_{i,<j},m\right)$ holds. Hence for some $n\in M_{i}$, $\psi\left(b_{i,j},n,m\right)$ holds. Hence $\psi\left(a_{0},n,m\right)$ holds (as $b_{i,j}\equiv_{M_{i}}a_{0}$) and hence $\psi\left(a_{j},n,m\right)$ holds. ∎

Definition 6.8**.**

Suppose $M$ is a model and $k<\omega$. Say that a formula $\varphi\left(x,a\right)$ $k$-Kim-divides over $M$ if there is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence $\langle a_{i}:i<\omega\rangle$ over $M$ starting with $a_{0}=a$ such that $\{\varphi\left(x,a_{i}\right):i<\omega\}$ is $k$-inconsistent.

Remark 6.9*.*

There is a choice involved in defining $k$-Kim-dividing, since it is not known if, in an NSOP1 theory, a formula that $k$-divides with respect to some \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence will also $k$-divide along a Morley sequence in a global invariant type. The above definition differs from the one implicitly used in [KRS19], but in light of Corollary 5.4 this definition seems reasonably canonical, given that any sequence which is a witness to Kim-dividing over $M$ will be an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$ and hence $\varphi\left(x,a\right)$ $k$-Kim-divides over $M$ for some $k<\omega$ iff $\varphi\left(x,a\right)$ Kim-divides over $M$.

Proposition 6.10**.**

Suppose that $\langle M_{i}:i<\alpha\rangle$ is an increasing sequence of models of $T$ with union $M=\bigcup_{i<\alpha}M_{i}$. Let $\varphi\left(x,y\right)$ be some formula (over $\emptyset$) and $a\in\mathbb{M}^{y}$. Fix some $k<\omega$.

1. (1)

If $\varphi\left(x,a\right)$ Kim-divides over $M$ then $\varphi\left(x,a\right)$ Kim-divides over $M_{i}$ for some $i<\alpha$. 2. (2)

If $\varphi\left(x,a\right)$ $k$-Kim-divides over $M_{i}$ for all $i<\alpha$ then $\varphi\left(x,a\right)$ $k$-Kim-divides over $M$.

Proof.

Note that this proposition, once proved, is immediately also true when we allow parameters from $M$ inside $\varphi$, as long as we assume these parameters are from $M_{0}$, by adding constants to the language. As the statement is trivial when $\alpha$ is a successor, we may assume $\alpha$ is a limit ordinal.

(1) Suppose that $\varphi\left(x,a\right)$ does not Kim-divide over any $M_{i}$. For $i<\alpha$, let $\bar{b}_{i}=\langle b_{i,j}:j<\omega\rangle$ be an indiscernible heir sequence starting with $b_{i,0}=a$ over $M_{i}$ (such a sequence exists, by e.g., taking a coheir sequence in reverse). In particular, $\bar{b}$ is an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence by symmetry. By Corollary 5.2, $\{\varphi\left(x,b_{i,j}\right):j<\omega\}$ is consistent. Let $\mathcal{U}$ be an ultrafilter on $\alpha$, concentrating on end-segments of $\alpha$. Let $\bar{a}=\langle a_{j}:j<\omega\rangle$ be a $\mathcal{U}$-average of $\langle\bar{b}_{i}:i<\alpha\rangle$ over $M$. Then Lemma 6.7 and symmetry imply that $\bar{a}$ is a \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$, and by construction $\{\varphi\left(x,a_{j}\right):j<\omega\}$ is consistent. By Corollary 5.2, $\varphi\left(x,a\right)$ does not Kim-divide over $M$.

(2) Suppose that $\varphi\left(x,a\right)$ $k$-Kim-divides over $M_{i}$ for all $i<\alpha$. For $i<\alpha$ let $\bar{b}_{i}=\langle b_{i,j}:j<\omega\rangle$ be a \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M_{i}$ witnessing this, i.e., $\{\varphi\left(x,b_{i,j}\right):j<\omega\}$ is $k$-inconsistent and $b_{i,0}=a$. As above, we let $\mathcal{U}$ be an ultrafilter on $\alpha$, concentrating on end-segments, and let $\bar{a}=\langle a_{j}:j<\omega\rangle$ be a $\mathcal{U}$-average of $\langle\bar{b}_{i}:i<\alpha\rangle$ over $M$. Then $\overline{a}$ is an $M$-indiscernible sequence in $\text{tp}(a/M)$ such that $\{\varphi(x;a_{i}):i<\omega\}$ is $k$-inconsistent, so is it is enough to show that $\bar{a}$ is a \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M$. By symmetry it is enough to show that a_{<j}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}^{K}a_{j} for all $j<\omega$. Suppose this is not the case, i.e., $\models\psi\left(a_{<j},a_{j},m\right)$ for some $m\in M$ and $j<\omega$, where $\psi\left(z,y,w\right)$ is an $L$-formula and $\psi\left(z,a_{j},m\right)$ Kim-divides over $M$, so also $\psi\left(z,a,m\right)$ Kim-divides over $M$. Hence, for some $S\in\mathcal{U}$, $m\in M_{i}$ and $\models\psi\left(b_{i,<j},b_{i,j},m\right)$ for all $i\in S$. Let $\langle N_{i}:i<\beta\rangle$ be an increasing enumeration of $\langle M_{i}:i\in S\rangle$. By (1), applied to $\langle N_{i}:i<\beta\rangle$ and the formula $\psi\left(z,a,m\right)$, we have that $\psi(z,a,m)$ Kim-divides over $M_{i}$ for some $i\in S$. Hence also $\psi\left(z,b_{i,j},m\right)$ Kim-divides over $M_{i}$ (as $b_{i,j}\equiv_{M_{i}}a$), contradicting the fact that $\bar{b}_{i}$ is a \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequence over $M_{i}$. ∎

6.3. Reformulating the Kim-Pillay-style characterization

Our final application will be an easy corollary of witnessing for \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}-Morley sequences, allowing us to give a more satisfying formulation of the Kim-Pillay-style characterization of Kim-independence. In [CR16, Proposition 5.3], a Kim-Pillay-style criterion was given for NSOP1, consisting of 5 axioms for an abstract independence relation on subsets of the monster model. Later, it was shown in [KRS19, Theorem 9.1] that any independence relation \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}} satisfying these axioms must strengthen \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K} in the sense that whenever $M\models T$ and a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}b, then also a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b. In order to characterize \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}, it was necessary to add an additional axiom to the list called witnessing: if a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}_{M}b witnessed by $\varphi(x;b)$ and $(b_{i})_{i<\omega}$ is a Morley sequence over $M$ in a global $M$-invariant (or even $M$-finitely satisfiable) type extending $\text{tp}(b/M)$, then $\{\varphi(x;b_{i}):i<\omega\}$ is inconsistent. Though useful in practice, this is somewhat unsatisfying, as it requires reference to independence notions like invariance or finite satisfiability instead of a property intrinsic to \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}.

Theorem 6.11**.**

Assume there is an $\text{Aut}(\mathbb{M})$-invariant ternary relation \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}} on small subsets of the monster $\mathbb{M}\models T$ which satisfies the following properties, for an arbitrary $M\models T$ and arbitrary tuples from $\mathbb{M}$.

1. (1)

Strong finite character: if a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}_{M}b, then there is a formula $\varphi(x,b,m)\in\text{tp}(a/bM)$ such that for any $a^{\prime}\models\varphi(x,b,m)$, a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}_{M}b. 2. (2)

Existence over models: $M\models T$ implies a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}M for any $a$. 3. (3)

Monotonicity: aa^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}bb^{\prime} $\implies$ a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}b. 4. (4)

Symmetry: a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}b\iff b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}a. 5. (5)

The independence theorem: a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}b, a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}c, b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}c and $a\equiv_{M}a^{\prime}$ implies there is $a^{\prime\prime}$ with $a^{\prime\prime}\equiv_{Mb}a$, $a^{\prime\prime}\equiv_{Mc}a^{\prime}$ and a^{\prime\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}bc. 6. (6)

\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}-Morley sequences are witnesses: if $M\models T$ and $I=(b_{i})_{i<\omega}$ is an $M$-indiscernible sequence with $b_{0}=b$ satisfying b_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}b_{<i}, then whenever a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,}}_{M}b, there is $\varphi(x;m,b)\in\text{tp}(a/Mb)$ such that $\{\varphi(x;m,b_{i}):i<\omega\}$ is inconsistent.

Then $T$ is NSOP1 and \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}=\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K} over models, i.e. if $M\models T$, a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}b if and only if a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b.

Proof.

Because \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}} satisfies axioms (1) through (5), it follows that $T$ is NSOP1 and for any $M\models T$, if a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}b then a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b, by [KRS19, Theorem 9.1]. For the other direction, suppose a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b. Let $I=\langle b_{i}:i<\omega\rangle$ be an $M$-finitely satisfiable Morley sequence over $M$ with $b_{0}=b$. As a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K}_{M}b, we find $a^{\prime}\equiv_{Mb}a$ so that $I$ is $Ma^{\prime}$-indiscernible. By [CR16, Claim in proof of Proposition 5.3], any relation \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}} satisfying (1)–(4), we have c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{u}_{M}d implies c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}d. Therefore, the sequence $I$ is, in particular, an \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}-Morley sequence over $M$ and $a^{\prime}\models\bigcup_{i<\omega}p(x;b_{i})$ so a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}_{M}b by (6). ∎

Remark 6.12*.*

In any NSOP1 theory, \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}^{\!\!\!\!\hbox to0.0pt{\scriptscriptstyle\textnormal{}\hss}\,\,\,\,}}^{K} satisfies properties (1)–(6), by Fact 2.7 and Theorem 5.1, so the existence of such a relation characterizes NSOP1 theories.