We study expansions of NSOP 1 theories that preserve NSOP 1 . We prove that if T is a model complete NSOP 1 theory eliminating the quantifier ∃ ∞ , then the generic expansion of T by arbitrary constant, function, and relation symbols is still NSOP 1 . We give a detailed analysis of the special case of the theory of the generic L-structure, the model companion of the empty theory in an arbitrary language L. Under the same hypotheses, we show that T may be generically expanded to an NSOP 1 theory with built-in Skolem functions. In order to obtain these results, we establish strengthenings of several properties of Kim-independence in NSOP 1 theories, adding instances of algebraic independence to their conclusions.Date: September 18, 2018.

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 ∀∃-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. 1The theory T m,n is also interesting from the perspective of the model theory of graphs. The class of graphs has a model companion, the unique countable model of which is known as the "random graph". This structure has a bipartite counterpart, the "random bipartite graph". When it is viewed as a structure in a language with unary predicates for the bipartition, the theory of the random bipartite graph is the model companion of the theory of arbitrary incidence structures. Both the random graph and the random bipartite graph have well-understood theories: they are ℵ 0 -categorical, simple, unstable, and have quantifier elimination and trivial algebraic closure.In the same way, T m,n can be viewed as the theory of existentially closed bipartite graphs omitting the complete bipartite graph K m,n . Thus the theories T m,n are bipartite counterparts to the theories T n of generic K n -free graphs (existentially closed graphs omitting a complete subgraph of size n). The theories T n , introduced by Henson [13], are important examples of countably categorical theories exhibiting the properties TP 2 , SOP 3 , and NSOP 4 (see [24], and also [9] for discussion of the model theoretic properties of Henson graphs). We show that for m, n ≥ 2, T m,n also has TP 2 , but it is NSOP 1 , hence tamer, in a sense, than the Henson graphs.In another contrast to the Henson graphs, T m,n is not countably categorical when m, n ≥ 2. In fact, we show that in this case, T m,n has continuum-many types over the empty set, and hence has no countable saturated model (Theorem 3.3). The question of whether T m,n has a prime model seems to be a hard combinatorial problem; in the case m = n = 2, we show that it is equivalent to a longstanding open problem in the theory of projective planes (Theorem 3.8).Finally, from the point of view of classification theory, the generic theory recipe has been a fruitful source of examples of simple theories (see [6], for example). Recently, many examples of generic theories which are not simple have been shown to be NSOP 1 (see [8], [19], [20]). The theories T m,n are further examples of this phenomen...

We define the interpolative fusion [Formula: see text] of a family [Formula: see text] of first-order theories over a common reduct [Formula: see text], a notion that generalizes many examples of random or generic structures in the model-theoretic literature. When each [Formula: see text] is model-complete, [Formula: see text] coincides with the model companion of [Formula: see text]. By obtaining sufficient conditions for the existence of [Formula: see text], we develop new tools to show that theories of interest have model companions.

Disjoint n-amalgamation is a condition on a complete first-order theory specifying that certain locally consistent families of types are also globally consistent. In this paper, we show that if a countably categorical theory T admits an expansion with disjoint n-amalgamation for all n, then T is pseudofinite. All theories which admit an expansion with disjoint n-amalgamation for all n are simple, but the method can be extended, using filtrations of Fraïssé classes, to show that certain non-simple theories are pseudofinite. As case studies, we examine two generic theories of equivalence relations, T * feq and T CPZ , and show that both are pseudofinite. The theories T * feq and T CPZ are not simple, but they have NSOP 1 . This is established here for T CPZ for the first time.

An automaton with advice is a finite state automaton which has access to an additional fixed infinite string called an advice tape. We refine the Myhill-Nerode theorem to characterize the languages of finite strings that are accepted by automata with advice. We do the same for tree automata with advice.