Given a set A ⊆ N, we consider the relationship between stability of the structure (Z, +, 0, A) and sparsity of the set A. We first show that a strong enough sparsity assumption on A yields stability of (Z, +, 0, A). Specifically, if there is a function f :). Such sets include examples considered by Palacín and Sklinos [19] and Poizat [23], many classical linear recurrence sequences (e.g. the Fibonaccci numbers), and any set in which the limit of ratios of consecutive elements diverges. Finally, we consider sparsity conclusions on sets A ⊆ N, which follow from model theoretic assumptions on (Z, +, 0, A). We use a result of Erdős, Nathanson, and Sárközy [8] to show that if (Z, +, 0, A) does not define the ordering on Z, then the lower asymptotic density of any finitary sumset of A is zero. Finally, in a theorem communicated to us by Goldbring, we use a result of Jin [11] to show that if (Z, +, 0, A) is stable, then the upper Banach density of any finitary sumset of A is zero.

In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of "generic stability" in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as φ-types in simple theories that are definable and finitely satisfiable in a small model, but not finitely approximated. Our proofs demonstrate interesting connections to classical results from Ramsey theory for finite graphs and hypergraphs.Date: May 28, 2019.

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We prove that, given ǫ > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A ⊆ G is k-stable. Then there is a normal subgroup H ≤ G of index at most n, and a set Y ⊆ G, which is a union of cosets of H, such that |A △ Y | ≤ ǫ|H|. It follows that, for any coset C of H, either |C ∩ A| ≤ ǫ|H| or |C \ A| ≤ ǫ|H|. This qualitatively generalizes recent work of Terry and Wolf on vector spaces over Fp.

We use axioms of abstract ternary relations to define the notion of a free amalgamation theory. These form a subclass of first-order theories, without the strict order property, encompassing many prominent examples of countable structures in relational languages, in which the class of algebraically closed substructures is closed under free amalgamation. We show that any free amalgamation theory has elimination of hyperimaginaries and weak elimination of imaginaries. With this result, we use several families of well-known homogeneous structures to give new examples of rosy theories. We then prove that, for free amalgamation theories, simplicity coincides with NTP 2 and, assuming modularity, with NSOP 3 as well. We also show that any simple free amalgamation theory is 1-based. Finally, we prove a combinatorial characterization of simplicity for Fraïssé limits with free amalgamation, which provides new context for the fact that the generic Kn-free graphs are SOP 3 , while the higher arity generic K r n -free r-hypergraphs are simple.