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Scott Mutchnik

Publications and source records attributed to Scott Mutchnik.

12 recordsLinked to original sources
arXiv2026

Reducing stable forking dependence to finitely many pregeometries

We show that one of the main cases of the stable forking conjecture, stability of the forking relation over a base in a finite-rank supersimple theory, is determined by finitely many pregeometries in each rank. This case of the stable forking conjecture has long had an implicitly well-known pregeometric interpretation: there is a set of matroids $\mathcal{G}_{n}$ such that the forking instability in rank $n$ is equivalent to the pregeometry on some rank-one partial type (over a finite set) embedding a matroid in $\mathcal{G}_{n}$. Our contribution is to show that this set of matroids $\mathcal{G}_{n}$, determining based forking stability in rank $n$, can be chosen to be finite. The main part of our proof was already accomplished in rank $3$ by Peretz, but does not extend as stated to higher ranks (and may or may not directly extend in a weaker sense to higher ranks, by shrinking terms). However, we obtain a sufficient substitute for Peretz's work in ranks $n > 3$: we turn Peretz's original universal result into an existence theorem. The rest of our proof refines an argument from multi-experiment parameter definability, originating from work in applied model theory by Li, Meshkat, Ovchinnikov, Pillay, Pogudin and Scanlon.

math.LO↗
arXiv2026

Some applications of the real strict order property hierarchy

We give applications of the properties $\mathrm{NSOP}_{r}$ for non-integer values of $r$ to problems on the original hierarchy $\mathrm{NSOP}_{n}$ for integer values of $n$. We first show that the properties $\mathrm{NSOP}_{r}$, previously defined for real values $r \geq 3$, are even well-defined for real values $r \geq 2$, showing that $\mathrm{NSOP}_{2} \subseteq \mathrm{NSOP}_{r}$ for our original definition of $\mathrm{NSOP}_{r}$ even when $2 < r < 3$. As a consequence, newness of all of the well-defined properties $\mathrm{NSOP}_{r}$ for non-integer $r$ would negatively resolve the problem of whether $\mathrm{NSOP}_{2}$ is equal to $\mathrm{NSOP}_{3}$. We then prove an approximate alternative between two possibilities: (1) that in extending Shelah's original $\mathrm{NSOP}_{n}$ hierarchy for integers $n \geq 3$ to the $\mathrm{NSOP}_{r}$ hierarchy for reals $r > 2$, we really did introduce new classification-theoretic properties, and (2) that $\mathrm{NSOP}_{n+1} \cap \mathrm{NTP}_{2} = \mathrm{NSOP}_{n} \cap \mathrm{NTP}_{2}$ for integers $n \geq 3$, which would resolve a central open problem in classification theory. More precisely, we give a rigorous sense in which (1) can fail on particularly general grounds, and then show that if (1) fails for these general reasons, (2) must be true. Finally, we apply cycle-removal techniques from the theory of the properties $\mathrm{NSOP}_{r}$ for real-values of $r$ to make progress on the question of whether $\mathrm{NSOP}_{2}$ is equal to $\mathrm{NSOP}_{3}$. We (a) show that if $\mathcal{H}$ is a hereditary class of structures defined by finitely many forbidden weakly embedded substructures, if every theory whose models have age $\mathcal{H}$ has $\mathrm{SOP}_{2}$, then every theory whose models have age $\mathcal{H}$ has $\mathrm{SOP}_{3}$, and (b) observe that we cannot replace $\mathrm{SOP}_{2}$ with $\mathrm{TP}$ here.

math.LO↗
arXiv2026

On the notion of a patterning property in model theory

Different kinds of definable patterns in the models of a first-order theory, such as the order property, the tree property, or the ($n$-)strict order property, allow us to distinguish theories according to their logical complexity. The complexity distinctions given by these definable patterns play a central role in model theory. However, a rigorous definition of the notion of a model-theoretic patterning property has yet to be established. We start by discussing different proposals from the literature for making the notion of a model-theoretic patterning property rigorous. Some examples will include the straight definability of Shelah, which will describe properties definable by a pattern of consistency and inconsistency in a formula and its negation, and the poset definability of Garcia and Mennuni, covering properties definable by interpreting a partial order embedding a given poset. We will also introduce a higher-arity version of straight definability. In our first main result, we will answer open questions of Bailetti and Garcia-Mennuni, showing that the $n$-strict order property $\mathrm{SOP}_{n}$ is straightly definable and poset definable even for integers $n \geq 4$. This will complete the categorization of all of the classical classification-theoretic properties as straightly definable. Our other main result will concern properties that are straightly definable without negation: the positively straightly definable properties defined by Bailetti. We will show using Saracino's theorem and results of Bodirsky, Bodor and Marimon that, in any countably categorical theory, implications between positively straightly definable properties must be exhibited at the level of $\exists\forall$-formulas. This will have special consequences under the assumption that $\mathrm{SOP}_{2}$ is equal to $\mathrm{SOP}_{3}$.

math.LO↗
arXiv2025

Approximations of the strict order property

We observe that the definition of Shelah's classical $\mathrm{NSOP}_{n}$ hierarchy for first-order theories, for integers $n \geq 3$, can be restated so that it extends to the case where $n$ is replaced with any real number $r \geq 3$. Using this observation, we define a potentially larger family of properties $\mathrm{NSOP}_{r}$ for real numbers $r \geq 3$. Motivated by the question of whether the integer-valued and real-valued hierarchies are distinct, we translate these hierarchies into the setting of hereditary classes, obtaining a new real-valued quantity of independent combinatorial interest, $\mathfrak{o}(\mathcal{H})$, associated with any hereditary class $\mathcal{H}$. We show that, when $\mathcal{H}$ is defined by a finite family of forbidden weakly embedded substructures, $\mathfrak{o}(\mathcal{H})$ is an integer. While Malliaris implicitly showed that the properties $\mathrm{NSOP}_{n}$ are equivalent to closure under helix maps between graphs, both our observation that the properties $\mathrm{NSOP}_{n}$ can be restated so that $n$ can be replaced with any real number at least $3$, and our result that $\mathfrak{o}(\mathcal{H})$ is an integer when $\mathcal{H}$ is a hereditary class defined by a finite family of forbidden weakly embedded substructures, are even exhibited by a special class of helix maps, the interval helix maps. These are helix maps that respect the direction of edges, and whose regions are disjoint unions of linearly ordered sets without any edges between them. Toward showing the conjectural claim that $\mathfrak{o}(\mathcal{H})$ is not an integer in general, and therefore that the real-valued $\mathrm{NSOP}_{r}$ hierarchy is distinct from the integer-valued $\mathrm{NSOP}_{n}$ hierarchy at the level of hereditary classes, we show that the statement that $\mathfrak{o}(\mathcal{H})$ is an integer in general cannot be exhibited by interval helix maps.

math.LO↗
arXiv2024

Simple Homogeneous Structures and Indiscernible Sequence Invariants

We introduce some properties describing dependence in indiscernible sequences: $F_{ind}$ and its dual $F_{Mb}$, the definable Morley property, and $n$-resolvability. Applying these properties, we establish the following results: We show that the degree of nonminimality introduced by Freitag and Moosa, which is closely related to $F_{ind}$ (equal in $\mathrm{DCF}_{0}$), may take on any positive integer value in an $\omega$-stable theory, answering a question of Freitag, Jaoui, and Moosa. Proving a conjecture of Koponen, we show that every simple theory with quantifier elimination in a finite relational language has finite rank and is one-based. The arguments closely rely on finding types $q$ with $F_{Mb}(q) = \infty$, and on $n$-resolvability. We prove some variants of the simple Kim-forking conjecture, a generalization of the stable forking conjecture to $\mathrm{NSOP}_{1}$ theories. We show a global analogue of the simple Kim-forking conjecture with infinitely many variables holds in every $\mathrm{NSOP}_{1}$ theory, and show that Kim-forking with a realization of a type $p$ with $\mathrm{F}_{Mb}(p) < \infty$ satisfies a finite-variable version of this result. We then show, in a low $\mathrm{NSOP}_{1}$ theory or when $p$ is isolated, if $p \in S(C)$ has the definable Morley property for Kim-independence, Kim-forking with realizations of $p$ gives a nontrivial instance of the simple Kim-forking conjecture itself. In particular, when $F_{Mb}(p) < \infty$ and $|S^{F_{Mb}(p) + 1}(C)| < \infty$, Kim-forking with realizations of $p$ gives us a nontrivial instance of the simple Kim-forking conjecture. We show that the quantity $F_{Mb}$, motivated in simple and $\mathrm{NSOP}_{1}$ theories by the above results, is in fact nontrivial even in stable theories.

math.LO↗
arXiv2023

Properties of independence in $\mathrm{NSOP}_3$ theories

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.

math.LO↗
arXiv2023

On the properties $\mathrm{SOP}_{2^{n+1}+1}$

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 $0$-$\eth$-independence denote forking-independence. Inductively, a formula $(n+1)$-$\eth$-divides over $M$ if it divides by every $n$-$\eth$-independent Morley sequence over $M$, and $(n+1)$-$\eth$-forks over $M$ if it implies a disjunction of formulas that $(n+1)$-$\eth$-divide over $M$; the associated independence relation over models is called $(n+1)$-$\eth$-independence. We show that a theory where $n$-$\eth$-independence is symmetric or transitive must be $\mathrm{NSOP}_{2^{n+1}+1}$. We then show that, in the classical examples of $\mathrm{NSOP}_{2^{n+1}+1}$ theories, $n$-$\eth$-independence is symmetric and transitive; in particular, there are strictly $\mathrm{NSOP}_{2^{n+1}+1}$ theories where $n$-$\eth$-independence is symmetric and transitive, leaving open the question of whether symmetry or transitivity of $n$-$\eth$-independence is equivalent to $\mathrm{NSOP}_{2^{n+1}+1}$.

math.LO↗
arXiv2022

Generic expansions and the group configuration theorem

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.

math.LO↗
arXiv2022

Conant-independence and generalized free amalgamation

We initiate the study of a generalization of Kim-independence, Conant-independence, based on the "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 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 Conant and Kruckman 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.

math.LO↗
arXiv2022

On NSOP$_2$ Theories

Answering a question of D\v{z}amonja and Shelah, we show that every NSOP$_2$ theory is NSOP$_1$.

math.LO↗