UC Berkeley

We discuss the classification of unstable theories in first-order logic.

In chapter 1, we initiate the study of a generalization of Kim-independence, \textit{Conant-independence}, based on the notion of \textit{strong Kim-dividing} of Kaplan, Ramsey and Shelah. We introduce an axiom on stationary independence relations essentially generalizing the ``freedom" axiom in some of the \textit{free amalgamation theories} of Conant, and show that this axiom provides the correct setting for carrying out arguments of Chernikov, Kaplan and Ramsey on $\mathrm{NSOP}_{1}$ theories relative to a stationary independence relation. Generalizing Conant's results on free amalgamation to the limits of our knowledge of the $\mathrm{NSOP}_{n}$ hierarchy, we show using methods from Conant as well as our previous work that any theory where the equivalent conditions of this local variant of $\mathrm{NSOP}_{1}$ holds is either $\mathrm{NSOP}_{1}$ or $\mathrm{SOP}_{3}$ and is either simple or $\mathrm{TP}_{2}$, and observe that these theories give an interesting class of examples of theories where Conant-independence is symmetric, including all of Conant's examples, the small cycle-free random graphs of Shelah and the (finite-language) $\omega$-categorical Hrushovski constructions of Evans and Wong.

We then answer a question of Conant, showing that the generic functional structures of Kruckman and Ramsey are examples of non-modular free amalgamation theories, and show that any free amalgamation theory is $\mathrm{NSOP}_{1}$ or $\mathrm{SOP}_{3}$, while an $\mathrm{NSOP}_{1}$ free amalgamation theory is simple if and only if it is modular.

Finally, we show that every theory where Conant-independence is symmetric is $\mathrm{NSOP}_{4}$. Therefore, symmetry for Conant-independence gives the next known neostability-theoretic dividing line on the $\mathrm{NSOP}_{n}$ hierarchy beyond $\mathrm{NSOP}_{1}$. We explain the connection to some established open questions.

\:In chapter 2, we exhibit a connection between geometric stability theory and the classification of unstable structures at the level of simplicity and the $\mathrm{NSOP}_{1}$-$\mathrm{SOP}_{3}$ gap. Particularly, we introduce generic expansions $T^{R}$ of a theory $T$ associated with a definable relation $R$ of $T$, which can consist of adding a new unary predicate or a new equivalence relation. When $T$ is weakly minimal and $R$ is a ternary fiber algebraic relation, we show that $T^{R}$ is a well-defined $\mathrm{NSOP}_{4}$ theory, and use one of the main results of geometric stability theory, the \textit{group configuration theorem} of Hrushovski, to give an exact correspondence between the geometry of $R$ and the classification-theoretic complexity of $T^{R}$. Namely, $T^{R}$ is $\mathrm{SOP}_{3}$, and $\mathrm{TP}_{2}$ exactly when $R$ is geometrically equivalent to the graph of a type-definable group operation; otherwise, $T^{R}$ is either simple (in the predicate version of $T^{R}$) or $\mathrm{NSOP}_{1}$ (in the equivalence relation version.) This gives us new examples of strictly $\mathrm{NSOP}_{1}$ theories.

\:In chapter 3, we prove the following fact:

\:$\mathrm{NSOP}_1$ is equal to $\mathrm{NSOP}_{2}$.

\:This answers an open question, first formally posed by Džamonja and Shelah in 2004, but attested in notes of Shelah based on lectures delivered at Rutgers University in fall of 1997.

\:In chapter 4, we prove some results about the theory of independence in $\mathrm{NSOP}_{3}$ theories that do not hold in $\mathrm{NSOP}_{4}$ theories. We generalize Chernikov's work on simple and co-simple types in $\mathrm{NTP}_{2}$ theories to types with $\mathrm{NSOP}_{1}$ induced structure in $\mathrm{N}$-$\omega$-$\mathrm{DCTP}_{2}$ and $\mathrm{NSOP}_{3}$ theories, and give an interpretation of our arguments and those of Chernikov in terms of the characteristic sequences introduced by Malliaris. We then prove an extension of the independence theorem to types in $\mathrm{NSOP}_{3}$ theories whose internal structure is $\mathrm{NSOP}_{1}$. Additionally, we show that in $\mathrm{NSOP}_{3}$ theories with symmetric Conant-independence, finitely satisfiable types satisfy an independence theorem similar to one conjectured by Simon for invariant types in $\mathrm{NTP}_{2}$ theories, and give generalizations of this result to invariant and Kim-nonforking types.

\:In chapter 5, we show that approximations of strict order can calibrate the fine structure of genericity. Particularly, we find exponential behavior within the $\mathrm{NSOP}_{n}$ hierarchy from model theory. Let $\ind^{\eth^{0}}$ denote forking-independence. Inductively, a formula \textit{$(n+1)$-$\eth$-divides} over $M$ if it divides by every $\ind^{\eth^{n}}$-Morley sequence over $M$, and \textit{$(n+1)$-$\eth$-forks} over $M$ if it implies a disjunction of formulas that $(n+1)$-$\eth$-divide over $M$; the associated independence relation over models is denoted $\ind^{\eth^{n+1}}$. We show that a theory where $\ind^{\eth^n}$ is symmetric must be $\mathrm{NSOP}_{2^{n+1}+1}$. We then show that, in the classical examples of $\mathrm{NSOP}_{2^{n+1}+1}$ theories, $\ind^{\eth^{n}}$ is symmetric and transitive; in particular, there are strictly $\mathrm{NSOP}_{2^{n+1}+1}$ theories where $\ind^{\eth^{n}}$ is symmetric and transitive, leaving open the question of whether symmetry or transitivity of $\ind^{\eth^{n}}$ is equivalent to $\mathrm{NSOP}_{2^{n+1}+1}$.