Abstract A first-order expansion of $(\mathbb {R},+,<)$ is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, p -adic fields, ordered abelian groups with only finitely many convex subgroups (in particular archimedean ordered abelian groups), and abelian groups equipped with archimedean cyclic group orders. The latter allows us to describe unary definable sets in dp-minimal expansions of $(\mathbb {Z},+,S)$ , where S is a cyclic group order. Along the way we describe unary definable sets in dp-minimal expansions of ordered abelian groups. In the last section we give a canonical correspondence between dp-minimal expansions of $(\mathbb {Q},+,<)$ and o-minimal expansions ${\mathscr R}$ of $(\mathbb {R},+,<)$ such that $({\mathscr R},\mathbb {Q})$ is a “dense pair.”.

We establish a list of characterizations of bounded twin-width for hereditary classes of totally ordered graphs: as classes of at most exponential growth studied in enumerative combinatorics, as monadically NIP classes studied in model theory, as classes that do not transduce the class of all graphs studied in finite model theory, and as classes for which model checking first-order logic is fixed-parameter tractable studied in algorithmic graph theory. This has several consequences. First, it allows us to show that every hereditary class of ordered graphs either has at most exponential growth, or has at least factorial growth. This settles a question first asked by Balogh et al. [ 5 ] on the growth of hereditary classes of ordered graphs, generalizing the Stanley-Wilf conjecture/Marcus-Tardos theorem. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width. Finally, it settles the small conjecture [ 8 ] in the case of ordered graphs.

An interpretation is an operation that maps an input graph to an output graph by redefining its edge relation using a first-order formula. This rich framework includes operations such as taking the complement or a fixed power of a graph as (very) special cases.

We establish a list of characterizations of bounded twin-width for hereditary classes of totally ordered graphs: as classes of at most exponential growth studied in enumerative combinatorics, as monadically NIP classes studied in model theory, as classes that do not transduce the class of all graphs studied in finite model theory, and as classes for which model checking first-order logic is fixed-parameter tractable studied in algorithmic graph theory.

Abstract We prove that every ultraproduct of p-adics is inp-minimal (i.e., of burden 1). More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic 0 in the RV language.

Abstract We show that dp-minimal valued fields are henselian and give classifications of dp-minimal ordered abelian groups and dp-minimal ordered fields without additional structure.

Abstract Let T be an NIP ${\cal L}$ -theory and $\mathop T\limits^\~ $ be an enrichment. We give a sufficient condition on $\mathop T\limits^\~$ for the underlying ${\cal L}$ -type of any definable (respectively invariant) type over a model of $\mathop T\limits^\~$ to be definable (respectively invariant). These results are then applied to Scanlon’s model completion of valued differential fields.

Abstract This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the (p,q)-theorem holds in dp-minimal theories of small or medium directionality. In the second part, we study dp-rank in dp-minimal theories and show that it enjoys many nice properties. It is continuous, definable in families and it can be characterised geometrically with no mention of indiscernible sequences. In particular, if the structure expands a divisible ordered abelian group, then dp-rank coincides with the dimension coming from the order.

Abstract We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst nonforking types.

The existence of a compression scheme for every concept class with bounded VC-dimension is one of the oldest open problems in statistical learning theory. Here we demonstrate the existence of such compression schemes under stronger assumptions than finite VC-dimension. Specifically, for each concept class we associate a family of concept classes that we call the alternating concept classes. Under the assumption that these concept classes have bounded VC dimension, we prove existence of a compression scheme. This result is motivated by recent progress in the field of model theory with respect to an analogues problem. In fact, our proof can be considered as a constructive proof of these advancements. This means that we describe the reconstruction function explicitly. Not less important, the theorems and proofs we present are in purely combinatorial terms and are available to the reader who is unfamiliar with model theory. Also, using tools from model theory, we apply our results and prove existence of compression schemes in interesting cases, such as concept classes defined by hyperplanes, polynomials, exponentials, restricted analytic functions and compositions, additions and multiplications of all of the above.

Abstract We address the following question: Can we expand an NIP theory by adding a linear order such that the expansion is still NIP? Easily, if acl(A)=A for all A, then this is true. Otherwise, we give counterexamples. More precisely, there is a totally categorical theory for which every expansion by a linear order has IP. There is also an ω-stable NDOP theory for which every expansion by a linear order interprets pseudofinite arithmetic.

Abstract This work builds on previous papers by Hrushovski, Pillay and the author where Keisler measures over NIP theories are studied. We discuss two constructions for obtaining generically stable measures in this context. First, we show how to symmetrize an arbitrary invariant measure to obtain a generically stable one from it. Next, we show that suitable sigma-additive probability measures give rise to generically stable Keisler measures. Also included is a proof that generically stable measures over o-minimal theories and the p-adics are smooth.

Abstract We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has nonempty interior, and any theory of pure tree is dp-minimal.