Properties of independence in $\mathrm{NSOP}_{3}$ theories

A central program in pure model theory is to develop the theory of independence, which originated within the stable theories, beyond stability and simplicity. This has been successful for the original notion of forking-independence within $\mathrm{NTP}_{2}$ theories: for example, Chernikov and Kaplan, in [9], show that forking coincides with dividing in $\mathrm{NTP}_{2}$ theories; Ben-Yaacov and Chernikov, in [42], give an independence theorem for forking-independence in $\mathrm{NTP}_{2}$ theories that is improved by Simon in [40], and Chernikov, in [8], studies simple types in $\mathrm{NTP}_{2}$ theories and gives a characterization of $\mathrm{NTP}_{2}$ theories in terms of Kim’s lemma. In a different direction, Kaplan and Ramsey in [19] extend the original theory of independence in simple theories to $\mathrm{NSOP}_{1}$ theories by introducing the notion of Kim-independence, described as forking-independence “at a generic scale.” Kaplan and Ramsey, in [19], show, using work of Chernikov and Ramsey in [10], that symmetry of Kim-independence characterizes the property $\mathrm{NSOP}_{1}$; they also show that the independence theorem for Kim-independence characterizes $\mathrm{NSOP}_{1}$. To give examples of further consequences of $\mathrm{NSOP}_{1}$ for the theory of Kim-independence, Kaplan and Ramsey in [20] give a characterization of $\mathrm{NSOP}_{1}$ in terms of transitivity, Kaplan, Ramsey and Shelah in [21] give a characterization in terms of local character; Dobrowolski, Kim and Ramsey in [13] and Chernikov, Kim and Ramsey in [6] study independence over arbitrary sets in $\mathrm{NSOP}_{1}$ theories. Kruckman and Ramsey, in [27], prove an improved independence theorem, developed further by Kruckman, Tran and Walsberg in the appendix of [28]. Kim ([23]) initiates a theory of canonical bases. For extensions to positive logic, see [12], [17], [5]; see also [7] for extensions of Kim-independence to $\mathrm{NTP}_{2}$ theories. Beyond $\mathrm{NSOP}_{1}$ and $\mathrm{NSOP}_{2}$, the author in [35] develops a theory of independence in $\mathrm{NSOP}_{2}$ theories and uses this to show that every $\mathrm{NSOP}_{2}$ theory is in fact $\mathrm{NSOP}_{1}$, and Kim and Lee, in [25], use remarks by the author in [35] to develop Kim-forking and Kim-dividing in the $\mathrm{NATP}$ theories introduced by Ahn and Kim in [3] and further devloped by Ahn, Kim and Lee in [4], as well as the related $\mathrm{N}$-$k$-$\mathrm{DCTP}_{2}$ theories introduced by the author in [35].

However, much remains to be understood about the theory of independence in Shelah’s strong order hierarchy, $\mathrm{NSOP}_{n}$, for $n\geq 3$. In [36], the author relativizes the theory of Kim-indpendence in [10], [19] by developing a theory of independence relative to abstract independence relations generalizing the free amalgamation axioms of [11]; though the theories to which this result applies may be strictly $\mathrm{NSOP}_{4}$ ($\mathrm{NSOP}_{4}$ and $\mathrm{SOP}_{3}$) as well as $\mathrm{NSOP}_{1}$, $\mathrm{NSOP}_{4}$ is not actually used in the result. The author also observes in the same paper using the generalization in [35] of the arguments of [11] that theories possessing independence properties with no known $\mathrm{NSOP}_{4}$ counterxamples–symmetric Conant-independence and the strong witnessing property that generalizes Kim’s lemma–cannot be strictly $\mathrm{NSOP}_{3}$. Conant-independence, which can be described as forking-independence at a maximally generic scale and is grounded in the strong Kim-dividing of [21], is introduced in that paper (based on a similar notion with the same name developed in [35] to show the equivalence of $\mathrm{NSOP}_{1}$ and $\mathrm{NSOP}_{2}$) as a potential extension of the theory of Kim-independence beyond $\mathrm{NSOP}_{1}$. There the author shows that a theory where Conant-independence is symmetric must be $\mathrm{NSOP}_{4}$, and characterizes Conant-independence in most of the known examples of $\mathrm{NSOP}_{4}$ theories, where it is symmetric. This leaves open the question of whether Conant-independence is symmetric in any $\mathrm{NSOP}_{4}$ theory, a question intimately related to the question of whether any $\mathrm{NSOP}_{3}$ theory is $\mathrm{NSOP}_{2}$. In [18], Kaplan, Ramsey and Simon have recently shown that all binary $\mathrm{NSOP}_{3}$ theories are simple, by developing a theory of independence for a class of theories containing all binary theories. In [34] the author develops the independence relations $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{\eth^{n}}$, based on the same idea of forking-independence at a maximally generic scale, shows that any theory where $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{\eth^{n}}$ is symmetric must be $\mathrm{NSOP}_{2^{n+1}+1}$, and characterizes $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{\eth^{n}}$ in the classical examples of $\mathrm{NSOP}_{2^{n+1}+1}$ theories, leaving open the question of whether $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{\eth^{n}}$ is symmetric in any $\mathrm{NSOP}_{2^{n+1}+1}$ theory. (Demonstrating robustness of the result, the author proves a similar result for left and right transitivity.) In [32], Malliaris and Shelah initiate a structure theory for $\mathrm{NSOP}_{3}$ theories, though instead of a theory of independence along the lines of forking-independence or Kim-independence, they show symmetric inconsistency for higher formulas, a result on sequences of realizations of two invariant types yielding inconsistent instances of two formulas, rather than any kind of indiscernible sequence witnessing the dividing of a single formula. Malliaris, in [33], also investigates the graph-theoretic depth of independence in $\mathrm{NSOP}_{3}$ theories. The pressing question remains, for $n\geq 3$: using the assumption that $T$ is $\mathrm{NSOP}_{n}$ (and possibly some additional assumptions that are not already known to collapse $\mathrm{NSOP}_{n}$ into $\mathrm{NSOP}_{1}$), can we show any properties of $T$ that fit into the program of generalizing the properties of independence in stable or simple theories, as was done for $\mathrm{NSOP}_{1}$ and $\mathrm{NSOP}_{2}$ theories?

The aim of this paper is to show that this question is tractable for $\mathrm{NSOP}_{3}$ theories, whose equivalence with $\mathrm{NSOP}_{1}$ remains open. We prove three results on $\mathrm{NSOP}_{3}$ theories, two about the $\mathrm{NSOP}_{1}$ “building blocks” of $\mathrm{NSOP}_{3}$ theories and the independence relations between them in the global $\mathrm{NSOP}_{3}$ structure, and one about $\mathrm{NSOP}_{3}$ theories with symmetric Conant-independence. All three of these results truly use $\mathrm{NSOP}_{3}$ in that they fail when the assumption is relaxed to $\mathrm{NSOP}_{4}$ (and the first two results, though both concerning the $\mathrm{NSOP}_{1}$ local structure, involve separate uses of $\mathrm{NSOP}_{3}$ in a sense that will become apparent.) The first and third result will also appear similar to properties known or proposed for $\mathrm{NTP}_{2}$ theories, in contrast to the open question of whether $\mathrm{NTP}_{2}\cap\mathrm{NSOP}_{n}$ coincides with simplicity for $n\geq 3$, which would suggest that $\mathrm{NSOP}_{n}$ is much different from $\mathrm{NTP}_{2}$.

We give an outline of the paper.

In Section 3 we generalize work of Chernikov ([8]) on simple types in $\mathrm{NTP}_{2}$ theories. As the property $\mathrm{N}$-$\omega$-$\mathrm{DCTP}_{2}$ is a subclass of $\mathrm{NATP}$ which is one potential solution $X$ to [26]’s proposed analogy “simple : $\mathrm{NTP}_{2}$ :: $\mathrm{NSOP}_{1}$ : $X$,” ([35], [25]), it is to be expected that the analogous result for “$\mathrm{NSOP}_{1}$ types” holds for $\mathrm{N}$-$\omega$-$\mathrm{DCTP}_{2}$ theories. What is not predicted by this analogy is that the same result on $\mathrm{NSOP}_{1}$ local structure holds in $\mathrm{NSOP}_{3}$ theories. Instead of generalizing the definition of simple types, we introduce a definition schema for the internal properties of a (partial) type, which is more natural in that it refers to the global properties of a structure associated with that type. (We could also have generalized the defintion of simple types to $\mathrm{NSOP}_{1}$ and gotten the same conclusion; see Remark 3.6.) We show that just as Chernikov implicitly showed in ([8]) for internally simple types in $\mathrm{NTP}_{2}$ theories, the assumption of $\mathrm{NSOP}_{3}$ controls how internally $\mathrm{NSOP}_{1}$ types relate to the rest of the structure:

See Definitions 3.1 and 3.3. When $T$ is only assumed to be $\mathrm{NSOP}_{4}$, we give an internally simple type $p(x)$ for which this fails.

We then interpret the proof of this result as well the results of Chernikov in [8] (and their direct generalization to $\mathrm{N}$-$\omega$-$\mathrm{DCTP}_{2}$) in terms of the characteristic sequences introduced by Malliaris in [30] to relate “classification-theoretic properties” of a theory to the “graph-theoretic properties” of hypergraphs, and used by Malliaris in [31] to study Keisler’s order. Internally to a type $p(x)$, what the ambient theory perceives to be an instance of co-$\mathrm{NSOP}_{1}$ (an instance of $\mathrm{NSOP}_{1}$ with parameters realizing $p(x)$) is simply a definable hypergraph making no reference to consistency. Model-theoretic properties of a theory will give control of the graph-theoretic structure of hypergraphs definable in that theory, similarly to Shelah’s classic result that an definable bipartite graph with the order property in an $\mathrm{NSOP}$ theory must even have the independence property. Applied in the case where the model-theoretic properties, such as simplicity and $\mathrm{NSOP}_{1}$, are assumed of the internal structure on $p(x)$, this will illuminate the proof in [8] of co-simplicity in $\mathrm{NTP}_{2}$ theories and our proof of co-$\mathrm{NSOP}_{1}$ in $\mathrm{N}$-$\omega$-$\mathrm{DCTP}_{2}$ and $\mathrm{NSOP}_{3}$ theories.

In Section 4, we discuss how internally $\mathrm{NSOP}_{1}$ types interrelate within the ambient structure of a $\mathrm{NSOP}_{3}$ theory, showing that their behavior is similar to how they would interrelate in a globally $\mathrm{NSOP}_{1}$ theory. By the Kim-Pillay characterization of $\mathrm{NSOP}_{1}$, Theorem 9.1 of [19], for no reasonable notion of independence could a full independence theorem hold in an $\mathrm{SOP}_{1}$ (that is, non-$\mathrm{NSOP}_{1}$) theory. However, we prove an independence theorem between internally $\mathrm{NSOP}_{1}$ types in $\mathrm{NSOP}_{3}$ theories:

Here $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{K^{*}}$ is Conant-independence, Definition 2.3. Motivating this result, in an $\mathrm{NSOP}_{1}$ theory, Conant-independence coincides with Kim-independence, $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{K}$ and is symmetric; compare [19], Theorem 6.5, which characterizes $\mathrm{NSOP}_{1}$. (Between tuples of realizations of two co-$\mathrm{NSOP}_{1}$ types $p_{i},p_{j}$ it coincides with Kim-diving independence.) While in proving this result, we apply 1.1, it does not just follow from co-$\mathrm{NSOP}_{1}$: we exhibit internally stable types $p_{1},p_{2},p_{3}$ in an $\mathrm{NSOP}_{4}$ theory $T$ for which this fails. This independence theorem for internally $\mathrm{NSOP}_{1}$ types in $\mathrm{NSOP}_{3}$ theories is not only of interest to the program of extending the theory of independence beyond $\mathrm{NSOP}_{1}$ theories; it is also of interest to the question of whether $\mathrm{NSOP}_{3}$ coincides with $\mathrm{NSOP}_{2}=\mathrm{NSOP}_{1}$. One potential approach to building a strictly $\mathrm{NSOP}_{3}$ theory (that is, one that is $\mathrm{SOP}_{2}$) is by starting with $\mathrm{NSOP}_{2}$ structures and somehow combining them to obtain a failure of $\mathrm{NSOP}_{1}$ in the form of a failure of the independence theorem: this result says that it is impossible to obtain an $\mathrm{NSOP}_{3}$ theory from such a construction. It may be of interest to ask whether there is any connection between this result on stability-theoretic independence and Theorem 7.7 of [33], which concerns the graph-theoretic depth of independence in $\mathrm{NSOP}_{3}$ theories.

In section 5, we consider $\mathrm{NSOP}_{3}$ theories where Conant-independence is symmetric. It is natural to assume this, as there is no known $\mathrm{NSOP}_{4}$ theory where Conant-independence or Conant-dividing independence is not symmetric. Simon, in [40], proves an improved independence theorem for $\mathrm{NTP}_{2}$ theories, Fact 5.1, and poses an existence question, Question 5.2, for invariant types with the same Morley sequence in $\mathrm{NTP}_{2}$ theories; an independence theorem for forking-independence, for invariant types with the same Morley sequence in $\mathrm{NTP}_{2}$ theories, would follow from a positive answer to this question, by Simon’s result. In an $\mathrm{NSOP}_{3}$ theory with symmetric Conant-independence, we prove a similar independence theorem for Conant-independence between finitely satisfiable types with the same Morley sequence:

This fails when $T$ is the model companion of triangle-free graphs, which is $\mathrm{NSOP}_{4}$ with symmetric (indeed trivial) Conant-independence. We also give an extension of this result from finitely satisfiable types to Kim-nonforking types when Conant-dividing independence is symmetric, which has the advantage of exploiting the full force of symmetry for Conant-independence. While this result is again of interest to the question of extending the theory of independence beyond $\mathrm{NSOP}_{1}=\mathrm{NSOP}_{2}$, since there is precedent (see [35]) for using facts about independence to prove the equivalence of classification-theoretic dividing lines, it is also of interest to another open question, whether an $\mathrm{NSOP}_{3}$ theory with symmetric Conant-independence is $\mathrm{NSOP}_{1}$.

One final remark: in Theorem 3.15 of [36], a strategy was suggested for proving the equivalence of $\mathrm{NSOP}_{3}$ and $\mathrm{NSOP}_{2}$ by proving two facts that have no known $\mathrm{NSOP}_{4}$ counterexamples, symmetry for Conant-independence and the strong witnessing property, for all $\mathrm{NSOP}_{4}$ or even all $\mathrm{NSOP}_{3}$ theories. The results of this paper suggest a different approach, via finding properties of independence in $\mathrm{NSOP}_{3}$ theories that distinguish them from $\mathrm{NSOP}_{4}$ theories.

Notations are standard. We will need some basic defintions and facts about some standard relations between sets, as well as some facts about $\mathrm{NSOP}_{1}$ and $\mathrm{NSOP}_{3}$ theories.

Relations between sets

Adler, in [2], defines some properties of abstract ternary relations $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}B$ between sets. In our case, we will assume $M$ is a model, and we will only need to refer to a few of these properties by name:

Left 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$\hss}\kern 1.42882pt}}_{M}B$ and $A\subseteq C$, there is some $B^{\prime}\equiv_{A}B$ with $C\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}B^{\prime}$.

Right 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$\hss}\kern 1.42882pt}}_{M}B$ and $B\subseteq C$, there is some $A^{\prime}\equiv_{B}A$ with $A^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 1.42882pt}}_{M}C$.

Symmetry: 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$\hss}\kern 1.42882pt}}_{M}B$ then $B\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 1.42882pt}}_{M}A$.

Chain condition with respect to invariant Morley sequences: 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$\hss}\kern 1.42882pt}}_{M}B$ and $I=\{B_{i}\}_{i<\omega}$ is an invariant Morley sequence over $M$ (see below) with $B_{0}=B$, then there is $I^{\prime}\equiv_{MB}I$ indiscernible over $MA$ with $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}I^{\prime}$.

We will refer to various relations between sets. For the convenience of the reader, here is an index of the notation to be used. Kim-independence and Kim-dividing, as well as Conant-independence and Conant-dividing, will be defined later in this section.

$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}^{i}b$ if $\mathrm{tp}(a/Mb)$ extends to a global $M$-invariant type

$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 1.42882pt}}^{u}_{M}b$ if $\mathrm{tp}(a/Mb)$ extends to a global $M$-finitely satisfiable type

$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 1.42882pt}}^{f}_{M}b$ if $a$ is forking-independent from $b$ over $M$

$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{K}_{M}b$ if $a$ is Kim-independent from $b$ over $M$

$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{K^{*}}_{M}b$ if $a$ is Conant-independent from $b$ over $M$

$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 1.42882pt}}^{Kd}_{M}b$ if $\mathrm{tp}(a/Mb)$ contains no formulas Kim-dividing over $M$

$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{K^{*}d}_{M}b$ if $\mathrm{tp}(a/Mb)$ contains no formulas Conant-dividing over $M$.

We will use $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{K^{+}}$ 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$\hss}\kern 1.42882pt}}^{K^{+}u}$ as ad-hoc notations in proofs; these will be defined in the course of those proofs.

We give an overview of some basic definitions. A global type $p(x)$ is a complete type over the sufficiently saturated model $\mathbb{M}$. For $M\prec\mathbb{M}$, a global type $p(x)$ is invariant over $M$ if $\varphi(x,b)\in p(x)$ and $b^{\prime}\equiv_{M}b$ implies $\varphi(x,b^{\prime})\in p(x)$. One class of types invariant over $M$ is the class of types that are finitely satisfiable over $M$, meaning any formula in the type is satisfied by some element of $M$. We say an infinite sequence $\{b_{i}\}_{i\in I}$, is an invariant Morley sequence over $M$ (in the type $p(x)$) if there is a fixed global type $p(x)$ invariant over $M$ so that $b_{i}\models p(x)|_{M\{b_{j}\}_{j<i}}$ for $i\in I$. If $p(x)$ is finitely satisfiable over $M$, we say $\{b_{i}\}_{i\in I}$ is a coheir Morley sequence or finitely satisfiable Morley sequence over $M$. Invariant Morley sequences over $M$ are indiscernible over $M$, and the EM-type of an invariant Morley sequence over $M$ depends only on $p(x)$. For $p(x),q(y)$ $M$-invariant types, $p(x)\otimes q(y)$ is defined so that $ab\models p(x)\otimes q(y)|_{A}$ for $M\subseteq A$ when $b\models q(y)|_{A}$ and $a\models p(x)|_{Ab}$.

Both $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{i}$ 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$\hss}\kern 1.42882pt}}^{u}$ have right extension, but it is sometimes advantageous to work with coheir Morley sequences rather than general invariant Morley sequences 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$\hss}\kern 1.42882pt}}^{u}$ is also known to have left extensions.

Kim-indepedence and $\mathrm{NSOP}_{1}$

We assume knowledge of basic simplicity theory and the definition of forking-independence. An extension of the theory of independence from simple theories to $\mathrm{NSOP}_{1}$ theories was developed by Kaplan and Ramsey in [19], via the definition of Kim-independence

Kim-independence in $\mathrm{NSOP}_{1}$ theories behaves, in many ways, like forking-independence in simple theories.

The independence theorem for Kim-independence in $\mathrm{NSOP}_{1}$ theories generalizes that of [24] for simple theories, which in turn generalizes stationarity of forking-independence (the uniqueness of nonforking-extensions) in stable theories. Part of our argument for the results of section 4 will require re-proving the independence theorem in the context of co-$\mathrm{NSOP}_{1}$ types. For motivation, we give the original statement:

Conant-independence

Conant-independence was introduced in a modified form in [35] to show that $\mathrm{NSOP}_{2}$ theories were $\mathrm{NSOP}_{1}$. The standard version was defined in [36], based on Conant’s implcit use of the concept in [11] to classify modular free amalgamation theories. It was proposed by the author of this paper as an extension of Kim-independence beyond $\mathrm{NSOP}_{1}$ theories.

In [36] it is shown that if Conant-independence is symmetric in a theory $T$, $T$ is $\mathrm{NSOP}_{4}$. In the same paper, Conant-indepedence is characterized for most of the known examples of $\mathrm{NSOP}_{4}$ theories, where it is shown to be symmetric. It is open whether Conant-independence is symmetric in all $\mathrm{NSOP}_{4}$ theories, or even all $\mathrm{NSOP}_{3}$ theories. It is also open whether theories with symmetric Conant-independence display the classification-theoretic behavior characteristic of theories with a good notion of free amalgamation, first studied in [11] and later improved upon in [36]: either $\mathrm{NSOP}_{1}$ or $\mathrm{SOP}_{3}$, and either simple or $\mathrm{TP}_{2}$.

Classification theory

In this paper, we will be interested in $\mathrm{NSOP}_{3}$ theories and how they differ from $\mathrm{NSOP}_{4}$ theories:

We will need nothing about $\mathrm{NSOP}_{4}$, other than that the below counterexamples to our results on $\mathrm{NSOP}_{3}$ theories are $\mathrm{NSOP}_{4}$, because they are free amalgamation theories; see [11], Theorem 4.4. We will need the following syntactic fact about $\mathrm{NSOP}_{3}$, proven independently by Malliaris (Conclusion 6.15, [33]) and Conant ([11], Proposition 7.2 and proof of Theorem 7.17):

Finally $\mathrm{NTP}_{2}$ and $\mathrm{N}$-$\omega$-$\mathrm{DCTP}_{1}$ will play a secondary role in this paper, but we will discuss some results on these classes that motivate our main results on $\mathrm{NSOP}_{3}$ theories.

The class $\mathrm{NATP}$ was introduced in [3] and further developed in [4] as a generalization of $\mathrm{NTP}_{2}$; it has been proposed as one possible answer to a question of Kruckman [26], on what class can be viewed to generalize properties of $\mathrm{NIP}$ and $\mathrm{NSOP}_{1}$ theories the same way $\mathrm{NTP}_{2}$ theories generalize properties of $\mathrm{NIP}$ and simple theories. It is still open to what extent the analogy holds; for example, whether Kim-forking coincides with Kim-dividing in $\mathrm{NATP}$ theories, as forking coincides with dividing in $\mathrm{NTP}_{2}$ theories. However, for $\mathrm{N}$-$\omega$-$\mathrm{DCTP}_{2}$ theories, introduced in [35] and further developed in [25], the equivalence of Kim-forking and Kim-dividing was proven in [25] after being proven for coheir Kim-dividing and coheir Kim-forking in [35].

Simple types were defined in [16]; then co-simple and $\mathrm{NTP}_{2}$ types were defined in [8]. We define co-$\mathrm{NSOP}_{1}$ types and give some equivalent definitions, similarly to Definition 6.7 of [8]. (When clear from context, when $p(x)$ is an $n$-type we refer to $p(\mathbb{M}^{n})$ by $p(\mathbb{M})$.

While Chernikov ([19], Defintion 6.7) gives an additional characterization of co-simplicity in terms of symmetry for forking-independence, it requires additional elements of the base to belong to $p(\mathbb{M})$. Giving a characterization of co-$\mathrm{NSOP}_{1}$ types in terms of symmetry for Kim-independence would be more complicated, because defining Kim-independence over arbitrary sets, rather than models, requires additional considerations; see [19]. However, co-$\mathrm{NSOP}_{1}$ types over $M$ do have a symmetry property over $M$, which will be useful in the sequel: