Model theory of Steiner triple systems

TL;DR

This paper investigates the model-theoretic properties of the Fra"{sse9 limit of finite Steiner triple systems, establishing its theory as the model completion with various logical features.

Contribution

It proves that the theory of the Fra"{sse9 limit is the model completion of Steiner triple systems and explores its complex model-theoretic properties.

Findings

The theory is the model completion of Steiner triple systems.

It has quantifier elimination and TP2.

It is not small and has NSOP1.

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 Fra\"{\i}ss\'e 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.