Abstract In light of a gap found by Krupiński, we give a new proof of associativity for the Morley (or “nonforking”) product of invariant measures in NIP theories.

Abstract We study the theory T m,n of existentially closed incidence structures omitting the complete incidence structure K m,n , which can also be viewed as existentially closed K m,n -free bipartite graphs. In the case m = n = 2, this is the theory of existentially closed projective planes. We give an $\forall \exists$ -axiomatization of T m,n , show that T m,n does not have a countable saturated model when m , n ≥ 2, and show that the existence of a prime model for T 2,2 is equivalent to a longstanding open question about finite projective planes. Finally, we analyze model theoretic notions of complexity for T m,n . We show that T m,n is NSOP 1 , but not simple when m , n ≥ 2, and we show that T m,n has weak elimination of imaginaries but not full elimination of imaginaries. These results rely on combinatorial characterizations of various notions of independence, including algebraic independence, Kim independence, and forking independence.

Abstract We show that if a first-order structure ${\cal M}$ , with universe ℤ, is an expansion of (ℤ,+,0) and a reduct of (ℤ,+,<,0), then ${\cal M}$ must be interdefinable with (ℤ ,+,0) or (ℤ ,+,<,0).

Abstract 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 NTP2 and, assuming modularity, with NSOP3 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 SOP3, while the higher arity generic $K_n^r$ -free r-hypergraphs are simple.