Model theory of Steiner triple systems
Silvia Barbina, Enrique Casanovas
TL;DR
The paper identifies the model-theoretic structure of Steiner triple systems by treating STS as Steiner quasigroups and showing that the theory $T^ op_ ext{Sq}$ is the model completion of the theory of Steiner quasigroups. It proves $T^ op_ ext{Sq}$ is complete with quantifier elimination, not small, and analyzes its place in dividing lines by establishing $ ext{TP}_2$ and $ ext{NSOP}_1$ while achieving elimination of hyperimaginaries and weak elimination of imaginaries. The Fraïssé limit $M_{ ext{F}}$ is shown to be a prime model of $T^ op_ ext{Sq}$, with the Fraïssé limit embedding all finite STS and not being saturated. Key structural results include a precise description of algebraic closure, rank-based elimination of $ orall^ ext{ } ext{}$, and a robust independence calculus via free amalgamation, situating STS within contemporary model-theoretic taxonomy of incidence structures.
Abstract
A Steiner triple system is a set $S$ together with a collection $\mathcal{B}$ of subsets of $S$ of size 3 such that any two elements of $S$ belong to exactly one element of $\mathcal{B}$. It is well known that the class of finite Steiner triple systems has a Fraïssé limit $M_{\mathrm{F}}$. Here we show that the theory $T^\ast_\mathrm{Sq}$ of $M_{\mathrm{F}}$ is the model completion of the theory of Steiner triple systems. We also prove that $T^\ast_\mathrm{Sq}$ is not small and it has quantifier elimination, $\mathrm{TP}_2$, $\mathrm{NSOP}_1$, elimination of hyperimaginaries and weak elimination of imaginaries.