Independence in Generic Incidence Structures

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 ∀∃-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...

Forking, Imaginaries, and Other Features Of

“…The reverse inclusion being trivial, we conclude that ⌣ satisfies all the previous properties except Base Monotonicity. This is similar to the case of K n,m -free bipartite graphs [15,Remark 4.17…”

“…The first assertion holds because | w ⌣ satisfies | a ⌣ -amalgamation over algebraically closed sets (Theorem 2.4). The proof is a classical induction similar to the proof of Lemma 4.12 or [15,Proposition 4.11].…”

“…Various examples of strictly NSOP 1 theories appear since then. Among them are(1) Generic L-structure T ∅ L [25]; (2) Generic K n,m -free bipartite graphs [15]; and(3) omega-free PAC fields [9]. ACFG shares many features with those three archetypical examples.…”

Model theory of Steiner triple systems

“…The class of NSOP 1 theories has recently become the object of close scrutiny and new natural examples are being discovered -see [9], [15] and [16]. In this section we show that T * Sq is TP 2 and NSOP 1 , thus adding a further example of a TP 2 and NSOP 1 theory to those described in [13].…”

Divide and Conquer: Dividing Lines and Universality

“…That is, show that NSOP 2 is a successful dividing line. (Shelah, 1993b), T ceq (see Definition 9.15(4)) the generic binary function (Kruckman and Ramsey, 2018), certain fields (e.g., ω-free PAC fields (Chatzidakis, 2002;Chernikov and Ramsey, 2016)), vector spaces with a generic bi-linear form (Chernikov and Ramsey, 2016) and the generic projective plane (Conant and Kruckman, 2017). Another serious candidate for being a good test problem is: Definition 9.23.…”