On $\mathrm{NSOP}_{2}$ Theories

One of the most exciting areas of research in modern model theory is the classification along various dividing lines of non-simple but otherwise tame theories, especially $\mathrm{NSOP}_{n}$ theories for $1\leq n\leq 3$. The first two of these properties, introduced in [12], require the nonexistence of certain trees:

The property $\mathrm{NSOP}_{3}$ is introduced in [29] as part of a family of notions $\mathrm{NSOP}_{n}$ for $n\geq 3$:

In [30] it is shown that $T^{*}_{feq}$, the model companion of the theory of parametrized equivalence relations, is $\mathrm{NSOP}_{1}$ but not simple; a limited number of further examples have since been found by various authors. Yet the main problem, posed by Džamonja and Shelah in [12], has remained unsolved:

In this paper we answer the latter question in the positive:

One reason for the significance of this problem comes from Shelah and Usvyatsov’s proposal in [30] to characterize classes of theories both internally in terms of the structure of their sufficiently saturated models, and externally in terms of orders on theories. The $\mathrm{NSOP}_{2}$ theories have a deep external characterization: under the generalized continuum hypothesis, Džamonja and Shelah [12] show that maximality in the order $\lhd^{*}$, an order related to the Keisler order, implies a combinatorial property related to $\mathrm{SOP}_{2}$, which Shelah and Usvyatsov then show in [30] to be the same as $\mathrm{SOP}_{2}$; later, Malliaris and Shelah in [24] show the equivalence between $\mathrm{SOP}_{2}$ and $\lhd^{*}$-maximality under the generalized continuum hypothesis. On the other hand, $\mathrm{NSOP}_{1}$ theories can be characterized internally not only in terms of trees, but through the theory of independence, in analogy with stability theory. It is well known that simple theories are characterized as those theories where forking and dividing behave in certain ways as they do in stable theories; for example, symmetry of forking characterizes simple theories. In [16], Kaplan and Ramsey show that Kim-forking, or forking witnessed by invariant Morley sequences, is the correct way of extending the theory of forking to $\mathrm{NSOP}_{1}$ theories from simple theories. By relaxing the requirement of base monotonicity, they extend the Kim-Pillay characterization of simple theories in terms of the existence of abstract independence relations to $\mathrm{NSOP}_{1}$ theories, and, more concretely, characterize $\mathrm{NSOP}_{1}$ theories by the symmetry of Kim-independence, by the independence theorem for Kim-independence, and by a variant of Kim’s lemma in simple theories, asserting that Kim-dividing of a formula, rather than dividing, is witnessed by any invariant Morley sequence. Our result that $\mathrm{NSOP}_{1}$ theories coincide with $\mathrm{NSOP}_{2}$ theories therefore shows a surprising agreement between dividing lines related to Keisler’s order and dividing lines related to independence.

We outline the paper and give a word on the strategy for the proof. In section 3, we develop in general theories a version of a construction used by Chernikov and Kaplan in [6] to study forking and dividing in $\mathrm{NTP}_{2}$ theories. In [1], Adler initiated the study of abstract relations between sets in a model, generalizing some of the properties of forking-independence, coheirs, and other concrete relations from model theory, and provided a set of potential axioms for these relations¹¹1Other than Adler’s work in [1] and Conant’s work on free amalgamation theories in [8], an additional observation which ultimately led us to the proof of this result is found in [11], where d’Elbée proposes the problem of explaining the apparent ubiquity of additional independence relations with no known concrete model-theoretic independence relations in $\mathrm{NSOP}_{1}$ theories, such as strong independence existing alongside Kim-independence in the theory $\mathrm{ACFG}$ (introduced as part of a more general class in [13]) of algebraically closed fields with a generic additive subgroup. He observes that just as in the case of free amalgamation of generic functional structures in [22] or generic incidence structures in [10], these stronger independence relations can be used to prove the equivalence of forking and dividing for complete types in many known $\mathrm{NSOP}_{1}$ theories. Before proving Theorem 1.6, we gave some very weak axioms (including stationarity, a feature of the examples considered by [11]) for abstract relations between sets over a model, which appeared to be very common in $\mathrm{NSOP}$ theories including strictly $\mathrm{NSOP}_{1}$ theories and $\mathrm{NSOP}_{4}$ theories, and proved that theories with such a relation could not be $\mathrm{NSOP}_{2}$; instead of considering Morley sequences in canonical coheirs as in the below, we used $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$-independent sequences for the abstract relation $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$, in the sense of Definition 7.5 of [8]. Note also that the property quasi-strong finite character considered below is a property of the examples in [11].. We notice that the construction of Chernikov and Kaplan can be relativized to relations between sets satisfying certain axioms, obtaining new relations between sets from old ones, and iterate this construction to obtain a canonical class of coheirs in any theory.

In section 4, we study this canonical class of coheirs in $\mathrm{NSOP}_{2}$ theories. Before the development of Kaplan and Ramsey’s theory of Kim-independence in $\mathrm{NSOP}_{1}$ theories in [16], Chernikov [5] proposed finding a theory of independence for $\mathrm{NSOP}_{2}$ theories, and the proof of our main result comes from our efforts to answer this proposal. Just as in [6], Chernikov and Kaplan’s construction gives maximal classes in the dividing order of Ben Yaacov and Chernikov [33], we show that in $\mathrm{NSOP}_{2}$ theories our variant of this construction gives minimal classes in the restriction of this order to coheir Morley sequences, proving an analogue of Kim’s lemma. As a by-product of this construction, we also initiate the theory of independence in a class related to the $\mathrm{NATP}$ theories of Ahn and Kim [2], the study of which was further developed by Ahn, Kim and Lee in [3], showing that under this assumption Kim-forking and Kim-dividing coincide for coheir Morley sequences. (See [20] for the question of finding an analogue for $\mathrm{NSOP}_{1}$ theories of the role that $\mathrm{NTP}_{2}$ theories play relative to simple theories, and developing Kim-independence in that analogue; that Kim-forking coincides with Kim-dividing for coheir Morley sequences in a related class gives us preliminary evidence that $\mathrm{NATP}$ completes this analogy.)

In section 5, we investigate behavior similar to $\mathrm{NSOP}_{1}$ theories in $\mathrm{NSOP}_{2}$ theories. We introduce the notion of Conant-independence, which will generalize the relation $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}_{M}B$ defined by $\mathrm{acl}(MA)\cap\mathrm{acl}(MB)=M$ in the free amalgamation theories introduced by Conant [8] (based on concepts used to study the isometry groups of Urysohn spheres in [32]); see the following section. While it will end up coinciding with Kim-independence in our case, we studied a version of Conant-independence in a potentially strictly $\mathrm{NSOP}_{1}$, potentially $\mathrm{SOP}_{3}$ generalization of free amalgamation theories in [26]. Conant-independence in $\mathrm{NSOP}_{2}$ theories can be defined as Kim-independence relative to canonical Morley sequences, just as $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$ is Kim-independence relative to free amalgamation Morley sequences (as in lemma 7.7 of [8]); it can also be defined by forcing Kim’s lemma on Kim-independence, requiring a formula to divide with respect to every Morley sequence instead of just one, as suggested in tentative remarks of Kim in [17] in his discussion of strong dividing in subtle theories. We show that many of Ramsey and Kaplan’s arguments on Kim-independence in $\mathrm{NSOP}_{1}$ theories in [16] can be generalized to Conant-independence in $\mathrm{NSOP}_{2}$ theories, including a chain condition, symmetry and a weak independnece theorem. (But as is apparent in [8] and [26], similar behavior can occur in a $\mathrm{SOP}_{3}$ theory, which is why the following section is essential to the proof of our main result.)

In section 6, we conclude the proof of Theorem 1.6. One consequence of Conant’s free amalgamation axioms (say, the freedom, closure and stationarity axioms, in Defintion 2.1 in [8]) is the following:

Let $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ denote free amalgamation and $A_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 1.42882pt}}^{a}_{M}B$, $A_{2}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 1.42882pt}}^{a}_{M}C$, 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$\hss}\kern 1.42882pt}}_{M}C$ with $A_{1}\equiv_{M}A_{2}$. Then there is some $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}_{M}BC$ with $A\equiv_{MB}A_{1}$ and $A\equiv_{MC}A_{2}$.

We will have shown in the prior section that Conant-independence is symmetric, and that a similar fact holds, roughly, when replacing free amalgamation with canonical coheirs 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$\hss}\kern 1.42882pt}}^{a}$ with Conant-independence. Conant shows in [8] that modular free amalgamation theories must either be simple or $\mathrm{SOP}_{3}$ (see [14] for a related result on countably categorical Hrushovski constructions), starting with a failure of forking-independence to coincide with $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{a}$ (because forking-independence cannot be symmetric unless a theory is simple) and using the above fact to build up a configuration giving $\mathrm{SOP}_{3}$. Starting, analogously, with the assumption that an $\mathrm{NSOP}_{2}$ theory $T$ is $\mathrm{SOP}_{1}$, so Kim-dividing independence is not symmetric and therefore fails to coincide with Conant-independence, we simulate Conant’s construction of an instance of $\mathrm{SOP}_{3}$. In short, we show that a $\mathrm{NSOP}_{2}$ theory is either $\mathrm{NSOP}_{1}$ or $\mathrm{SOP}_{3}$. But a $\mathrm{NSOP}_{2}$ theory is of course not $\mathrm{SOP}_{3}$, so it must be $\mathrm{NSOP}_{1}$.

We let $a,b,c,d,e,A,B,C$ denote sets, potentially with an enumeration depending on context, and $x,y,z,X,Y,Z$ denote tuples of variables. We let $\mathbb{M}$ denote a sufficiently saturated model of a theory $T$ and let $M$ denote an elementary submodel. We write $AB$ to denote the union (or concatenation) of the sets $A$ and $B$, and write $I$, $J$, etc. for infinite sequences (or sometimes trees) of tuples or an infinite linearly ordered set.

Relations between sets

Roughly following the axioms for abstract independence relations in [1], as well as others that are standard in the literature, we define the following axioms for 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 over a model:

Invariance: For all $\sigma\in\mathrm{Aut}(\mathbb{M})$, $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 1.42882pt}}_{M}B$ implies $\sigma(A)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 1.42882pt}}_{\sigma(M)}\sigma(B)$.

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

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$\hss}\kern 1.42882pt}}_{M}C$.

Left monotonicity: If $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}B$ and $M\subseteq A^{\prime}\subseteq A$, then $A^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 1.42882pt}}_{M}B$

Right monotonicity: If $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}B$ and $M\subseteq B^{\prime}\subseteq B$, then $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}B^{\prime}$

(We will refer to the two previous properties, taken together, as monotonicity.)

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$\hss}\kern 1.42882pt}}_{M}A$

Coheirs and Morley sequences

A global type $p$ is a complete type over $\mathbb{M}$. For $p\in S(A)$ for $M\subseteq A$, we say $p$ is finitely satisfiable over $M$ or a coheir extension of its restriction to $M$ if every formula in $p$ is satisfiable in $M$. Global types $p$ finitely satisfiable in $M$ are invariant over $M$: whether $\varphi(x,b)$ belongs to $p$ for $\varphi$ a formula without parameters, depends only on the type of the parameter $b$ over $M$. 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$\hss}\kern 1.42882pt}}^{u}_{M}b$ to denote that $\mathrm{tp}(a/Mb)$ is finitely satisfiable in $M$. We let $a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{M}^{h}b$ denote $b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 1.42882pt}}^{u}_{M}a$. The relation $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{u}$ (over models) is well-known to satisfy all of the above properties other than symmetry. We say $\{b_{i}\}_{i\in I}$, for $I$ potentially finite, is a coheir sequence over $M$ if $b_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\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$\hss}\kern 1.42882pt}}^{u}_{M}b_{<i}$ for $i\in I$. We say a coheir sequence $\{b_{i}\}_{i\in I}$, for $I$ infinite, is moreover a coheir Morley sequence over $M$ if there is a fixed global type $p(x)$ finitely satisfiable in $M$ so that $b_{i}\models p(x)|_{\{Mb_{j}\}_{j<i}}$ for $i\in I$. The type of a coheir Morley sequence over $M$ (indexed by a given set) is well-known to depend only on $p(x)$, and coheir Morley sequences are known to be indiscernible; the type of a coheir sequence over $M$ depends only on the global coheirs over $M$ extending the $\mathrm{tp}(b_{i}/Mb_{<i})$.

$\mathrm{NSOP}_{1}$ theories and Kim-dividing

In this paper we use nonstandard terminology: Kim-dividing, etc. are defined in terms of Morley sequences in invariant types over $M$ rather than finitely satisfiable types over $M$ in [16]. The reason why we do this is 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$\hss}\kern 1.42882pt}}^{u}$ is known to satisfy left extension. This will do us no harm for our main result, though when we briefly consider Kim-forking in some $\mathrm{NATP}$ theories, we will note the nonstandard usage.

The following follows directly from Proposition 5.2 of [7]; see also Proposition 3.22 of [16] (where the evident argument for the version for invariant types is given) and Theorem 5.16 of [16] for the full symmetry characterization of $\mathrm{NSOP}_{1}$.

$\mathrm{NSOP}_{2}$ theories

A characterization of $\mathrm{SOP}_{2}$ as $k$-$\mathrm{TP}_{1}$ was proven by Kim and Kim in [18], where they also introduce the notion of weak $k$-$\mathrm{TP}_{1}$, prove that it implies $\mathrm{SOP}_{1}$, and conjecture that it also implies $\mathrm{SOP}_{2}$:

Later, Chernikov and Ramsey, in Theorem 4.8 of [7], claim to show that weak $k$-$\mathrm{TP}_{1}$ implies $\mathrm{SOP}_{2}$, but their proof is incorrect; the embedded tree $\{b_{\eta}\}_{\eta\in\omega^{<\omega}}$ in the proof of that theorem is not actually strongly indiscernible over the parameter set $C$. In an earlier version of this paper, we used this result. In this section, we will introduce an equivalent form of $\mathrm{SOP}_{2}$ that will suffice for our argument, and use the same method as [7] to give a proof that will work to show this equivalence despite failing for weak $k$-$\mathrm{TP}_{1}$.

So for example, all descending combs of length $n$ have the same quantifier-free type in the language $\{<_{\mathrm{lex}},\lhd,\wedge\}$ as the descending comb $\langle 0\rangle^{n-1}\smallfrown\langle 1\rangle,\ldots\langle 1\rangle$; meanwhile, $\langle 00\rangle,\langle 01\rangle,\langle 10\rangle,\langle 11\rangle$ is an example of a lexicographically ordered antichain that is not a descending comb.

Though the proof of Theorem 4.8 of [7] is incorrect, that theorem (albeit, not a “local” version) will be a corollary of our main result, Theorem 1.6, and the result of [18] that weak $k$-$\mathrm{TP}_{1}$ implies $\mathrm{SOP}_{1}$. (Note that $\mathrm{SOP}_{2}$ is just weak $2$-$\mathrm{TP}_{1}$).

The following section will require no assumptions on $T$. Iterating a similar construction to the one used by Chernikov and Kaplan in [6] to prove the equivalence of forking and dividing for formulas in $\mathrm{NTP}_{2}$ theories, we will contruct a canonical class of coheir extensions in any theory. This class will end up satisfying a variant of the “Kim’s lemma for Kim-dividing” in $\mathrm{NSOP}_{1}$ theories (Theorem 3.16 of [16]) when considered in a $\mathrm{NSOP}_{2}$ theory.

The “weakest” clause is not necessary for the main result, but we include it anyway to show our construction is canonical.

We start by relativizing the notions of Kim-dividing, Kim-forking, and quasi-dividing (Definition 3.2 of [6]) to an $M$-invariant ideal on the definable subsets of $\mathbb{M}$.

The proof of the following lemma is adapted straightforwardly from the proof of the “broom lemma” of Chernikov and Kaplan (Lemma 3.1 of [6])²²2Alex Kruckman, in a personal communication with the author, discussed an alternative to this proof for showing the properness of the ideal corresponding to the independence result 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$\hss}\kern 1.42882pt}}^{\mathrm{CK}}$, with the broom lemma as a corollary, which works for invariant Morley sequences as well as coheir Morley sequences; it is based on unpublished work of James Hanson on the concept of “fracturing,” a generalization of quasi-forking and quasi-dividing.. For the convenience of the reader we give a simplified proof of the modified version; note that this version is just a rephrasing in terms of ideals of Lemma 4.19 in [4]:

We now begin our construction. The following terminology comes from the notion of strong finite character (used in e.g. [7]).

We are now in a position to study derived independence relations:

The next observation is required to produce a relation with the coheir chain condition: