In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called "uniform definability of types over finite sets" (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.

In this paper, we introduce the notion of $${\textbf{K}} $$ -rank, where $${\textbf{K}} $$ is a strong amalgamation Fraïssé class. Roughly speaking, the $${\textbf{K}} $$ -rank of a partial type is the number “copies” of $${\textbf{K}} $$ that can be “independently coded” inside of the type. We study $${\textbf{K}} $$ -rank for specific examples of $${\textbf{K}} $$, including linear orders, equivalence relations, and graphs. We discuss the relationship of $${\textbf{K}} $$ -rank to other ranks in model theory, including dp-rank and (...) op-dimension (a notion coined by the first author and C. D. Hill in previous work). (shrink)

In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablity and show that this lies strictly between VC-minimality and dp-minimality. To do this we prove a general result about set systems with independence dimension ≤ 1. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, (...) show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density. (shrink)

We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraïssé classes and by complete prime filter classes. We exhibit the relationship between this and collapse-of-indiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.

In this paper, we study Vapnik‐Chervonenkis density (VC‐density) over indiscernible sequences (denoted VCind‐density). We answer an open question in [1], showing that VCind‐density is always integer valued. We also show that VCind‐density and dp‐rank coincide in the natural way.

In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.

We show that any formula with two free variables in a Vapnik–Chervonenkis (VC) minimal theory has VC-codensity at most 2. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where acleq= dcleq, the VC-codensity of a formula is at most the number of free variables (from the work of Aschenbrenner et al., the author, and Laskowski).