We define a new type of “shatter function” for set systems that satisfies a Sauer–Shelah type dichotomy, but whose polynomial-growth case is governed by Shelah’s two-rank instead of VC dimension. We identify the least exponent bounding the rate of growth of the shatter function, the quantity analogous to VC density, with Shelah’s $\omega $-rank.

A complete list of one dimensional groups definable in the p-adic numbers is given, up to a finite index subgroup and a quotient by a finite subgroup.

We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property. We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation subclasses whose axiomatisations are recursively enumerable in our second-order fragment can also be recursively axiomatised in their original first-order language. We pin down the expressive power of this formalism with respect to first-order logic, and investigate some questions relating to decidability and computational complexity. As applications of these results, by showing that certain classes can be straightforwardly defined as separation subclasses, we obtain first-order axiomatisability results for these classes. In particular we apply this technique to graph colourings and a class of partial algebras arising from separation logic.

We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property.

We build a new spectrum of recursive models ($ \operatorname {\mathrm {SRM}}(T)$) of a strongly minimal theory. This theory is non-disintegrated, flat, model complete, and in a language with a finite signature.

We prove that if an $\omega $-categorical structure has an $\omega $-categorical homogeneous Ramsey expansion, then so does its model-complete core.

We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have a generic pair of automorphisms.

We define the Scott complexity of a countable structure to be the least complexity of a Scott sentence for that structure. This is a finer notion of complexity than Scott rank: it distinguishes between whether the simplest Scott sentence is $\Sigma _{\alpha }$, $\Pi _{\alpha }$, or $\mathrm {d-}\Sigma _{\alpha }$. We give a complete classification of the possible Scott complexities, including an example of a structure whose simplest Scott sentence is $\Sigma _{\lambda + 1}$ for $\lambda $ a limit ordinal. This answers a question left open by A. Miller.

We also construct examples of computable structures of high Scott rank with Scott complexities $\Sigma _{\omega _1^{CK}+1}$ and $\mathrm {d-}\Sigma _{\omega _1^{CK}+1}$. There are three other possible Scott complexities for a computable structure of high Scott rank: $\Pi _{\omega _1^{CK}}$, $\Pi _{\omega _1^{CK}+1}$, $\Sigma _{\omega _1^{CK}+1}$. Examples of these were already known. Our examples are computable structures of Scott rank $\omega _1^{CK}+1$ which, after naming finitely many constants, have Scott rank $\omega _1^{CK}$. The existence of such structures was an open question.

Inquisitive modal logic, InqML, is a generalisation of standard Kripke-style modal logic. In its epistemic incarnation, it extends standard epistemic logic to capture not just the information that agents have, but also the questions that they are interested in. Technically, InqML fits within the family of logics based on team semantics. From a model-theoretic perspective, it takes us a step in the direction of monadic second-order logic, as inquisitive modal operators involve quantification over sets of worlds. We introduce and investigate the natural notion of bisimulation equivalence in the setting of InqML. We compare the expressiveness of InqML and first-order logic in the context of relational structures with two sorts, one for worlds and one for information states, and characterise inquisitive modal logic as the bisimulation invariant fragment of first-order logic over various natural classes of two-sorted structures.

We introduce the framework of AECats (abstract elementary categories), generalizing both the category of models of some first-order theory and the category of subsets of models. Any AEC and any compact abstract theory (“cat”, as introduced by Ben-Yaacov) forms an AECat. In particular, we find applications in positive logic and continuous logic: the category of (subsets of) models of a positive or continuous theory is an AECat. The Kim–Pillay theorem for first-order logic characterizes simple theories by the properties dividing independence has. We prove a version of the Kim–Pillay theorem for AECats with the amalgamation property, generalizing the first-order version and existing versions for positive logic.

Let $\mathcal M=(M,<,\ldots)$ be a linearly ordered first-order structure and T its complete theory. We investigate conditions for T that could guarantee that $\mathcal M$ is not much more complex than some colored orders (linear orders with added unary predicates). Motivated by Rubin’s work [5], we label three conditions expressing properties of types of T and/or automorphisms of models of T. We prove several results which indicate the “geometric” simplicity of definable sets in models of theories satisfying these conditions. For example, we prove that the strongest condition characterizes, up to definitional equivalence (inter-definability), theories of colored orders expanded by equivalence relations with convex classes.

We give an example of two ordered structures $\mathcal {M},\mathcal {N}$ in the same language $\mathcal {L}$ with the same universe, the same order and admitting the same one-variable definable subsets such that $\mathcal {M}$ is a model of the common theory of o-minimal $\mathcal {L}$-structures and $\mathcal {N}$ admits a definable, closed, bounded, and discrete subset and a definable injective self-mapping of that subset which is not surjective. This answers negatively two question by Schoutens; the first being whether there is an axiomatization of the common theory of o-minimal structures in a given language by conditions on one-variable definable sets alone. The second being whether definable completeness and type completeness imply the pigeonhole principle. It also partially answers a question by Fornasiero asking whether definable completeness of an expansion of a real closed field implies the pigeonhole principle.

We generalize a well-known theorem binding the elementary equivalence relation on the level of PAC fields and the isomorphism type of their absolute Galois groups. Our results concern two cases: saturated PAC structures and nonsaturated PAC structures.

We study computable Polish spaces and Polish groups up to homeomorphism. We prove a natural effective analogy of Stone duality, and we also develop an effective definability technique which works up to homeomorphism. As an application, we show that there is a $\Delta ^0_2$ Polish space not homeomorphic to a computable one. We apply our techniques to build, for any computable ordinal $\alpha $, an effectively closed set not homeomorphic to any $0^{(\alpha )}$-computable Polish space; this answers a question of Nies. We also prove analogous results for compact Polish groups and locally path-connected spaces.

A linear space is a system of points and lines such that any two distinct points determine a unique line; a Steiner k-system (for $k \geq 2$) is a linear space such that each line has size exactly k. Clearly, as a two-sorted structure, no linear space can be strongly minimal. We formulate linear spaces in a (bi-interpretable) vocabulary $\tau $ with a single ternary relation R. We prove that for every integer k there exist $2^{\aleph _0}$-many integer valued functions $\mu $ such that each $\mu $ determines a distinct strongly minimal Steiner k-system $\mathcal {G}_\mu $, whose algebraic closure geometry has all the properties of the ab initio Hrushovski construction. Thus each is a counterexample to the Zilber Trichotomy Conjecture.

Henle, Mathias, and Woodin proved in [21] that, provided that ${\omega }{\rightarrow }({\omega })^{{\omega }}$ holds in a model M of ZF, then forcing with $([{\omega }]^{{\omega }},{\subseteq }^*)$ over M adds no new sets of ordinals, thus earning the name a “barren” extension. Moreover, under an additional assumption, they proved that this generic extension preserves all strong partition cardinals. This forcing thus produces a model $M[\mathcal {U}]$, where $\mathcal {U}$ is a Ramsey ultrafilter, with many properties of the original model M. This begged the question of how important the Ramseyness of $\mathcal {U}$ is for these results. In this paper, we show that several classes of $\sigma $-closed forcings which generate non-Ramsey ultrafilters have the same properties. Such ultrafilters include Milliken–Taylor ultrafilters, a class of rapid p-points of Laflamme, k-arrow p-points of Baumgartner and Taylor, and extensions to a class of ultrafilters constructed by Dobrinen, Mijares, and Trujillo. Furthermore, the class of Boolean algebras $\mathcal {P}({\omega }^{{\alpha }})/{\mathrm {Fin}}^{\otimes {\alpha }}$, $2\le {\alpha }<{\omega }_1$, forcing non-p-points also produce barren extensions.

We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with signature $(0,1,+,\cdot )$. In the characteristic zero case, the claim implies that the full theory is undecidable, for rings of the referred type. This extends a series of results by Raphael Robinson, holding for certain polynomial integral domains, to a more general class.

We set up a general context in which one can prove Sauer–Shelah type lemmas. We apply our general results to answer a question of Bhaskar [1] and give a slight improvement to a result of Malliaris and Terry [7]. We also prove a new Sauer–Shelah type lemma in the context of $ \operatorname {\textrm{op}}$-rank, a notion of Guingona and Hill [4].

We consider the structures $(\mathbb {Z}; \mathrm {SF}^{\mathbb {Z}})$, $(\mathbb {Z}; <, \mathrm {SF}^{\mathbb {Z}})$, $(\mathbb {Q}; \mathrm {SF}^{\mathbb {Q}})$, and $(\mathbb {Q}; <, \mathrm {SF}^{\mathbb {Q}})$ where $\mathbb {Z}$ is the additive group of integers, $\mathrm {SF}^{\mathbb {Z}}$ is the set of $a \in \mathbb {Z}$ such that $v_{p}(a) < 2$ for  every prime p and corresponding p-adic valuation $v_{p}$, $\mathbb {Q}$ and $\mathrm {SF}^{\mathbb {Q}}$ are defined likewise for rational numbers, and $<$ denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences.

Drawing on the analogy between any unary first-order quantifier and a “face operator,” this paper establishes several connections between model theory and homotopy theory. The concept of simplicial set is brought into play to describe the formulae of any first-order language L, the definable subsets of any L-structure, as well as the type spaces of any theory expressed in L. An adjunction result is then proved between the category of o-minimal structures and a subcategory of the category of linearly ordered simplicial sets with distinguished vertices.