While finitely generic (f.g.) dimension groups are known to admit no proper self-embeddings, these groups also have no automorphisms other than scalar multiplications, and every countable infinitely generic (i.g.) dimension group admits proper self-embeddings and has automorphisms other than scalar multiplications. The finite-forcing companion of the theory of dimension groups is recursively isomorphic to first-order arithmetic, the infinite-forcing companion of the theory of dimension groups is recursively isomorphic to second-order arithmetic, and the first-order theory of existentially closed (e.c.) dimension groups is a complete ${\Pi }_1^1$-set. While many special properties of f.g. dimension groups may be realized in recursive e.c. dimension groups, and many special properties of i.g. dimension groups may be realized in hyperarithmetic e.c. dimension groups, no f.g. dimension group is arithmetic and no i.g. dimension group is analytical. Yet there is an f.g. dimension group recursive in first-order arithmetic, and (modulo a set-theoretic hypothesis) there is an i.g. dimension group recursive in second-order arithmetic.

We study semi-isolation as a binary relation on the locus of a complete type and prove that—under some additional assumptions—it induces the strict order property.

Suppose that $\sigma \in {\mathcal{L}}_{{\omega }_1,\omega }\left(\mathrm{L}\right)$ is such that all equations occurring in $\sigma$ are positive, have the same set of variables on each side of the equality symbol, and have at least one function symbol on each side of the equality symbol. We show that $\sigma$ satisfies Vaught’s conjecture. In particular, this proves Vaught’s conjecture for sentences of ${\mathcal{L}}_{{\omega }_1,\omega }\left(\mathrm{L}\right)$ without equality.

This paper presents some finite combinatorics of set systems with applications to model theory, particularly the study of dependent theories. There are two main results. First, we give a way of producing lower bounds on ${\mathrm{VC}}_{\mathrm{ind}}$-density and use it to compute the exact ${\mathrm{VC}}_{\mathrm{ind}}$-density of polynomial inequalities and a variety of geometric set families. The main technical tool used is the notion of a maximum set system, which we juxtapose to indiscernibles. In the second part of the paper we give a maximum set system analogue to Shelah’s characterization of stability using indiscernible sequences.

Answering a question in the reverse mathematics of combinatorial principles, we prove that the thin set theorem for pairs (TS(2)) implies the diagonally noncomputable set principle (DNR) over the base axiom system ${\mathrm{RCA}}_0$.

This paper introduces a new theory of constant regions, which generalizes that of interstices, in nonstandard models of arithmetic. In particular, we show that two homogeneity notions introduced by Richard Kaye and the author, namely, constantness and pregenericity, are equivalent. This led to some new characterizations of generic cuts in terms of existential closedness.