Independence in generic incidence structures

TL;DR

This paper investigates the model theory of existentially closed incidence structures omitting complete bipartite graphs, providing axiomatizations, analyzing model-theoretic properties, and connecting to open problems in finite geometry.

Contribution

It introduces an axiomatization of $T_{m,n}$, explores its model-theoretic complexity, and links the existence of prime models to open questions in finite projective planes.

Findings

$T_{m,n}$ is NSOP$_1$ but not simple for $m,n ext{geq} 2$

No countable saturated models exist for $T_{m,n}$ when $m,n ext{geq} 2$

Existence of a prime model for $T_{2,2}$ relates to a longstanding open problem

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 $m,n\geq 2$, 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 $m,n\geq 2$, 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.