Model theory of Steiner triple systems

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\"ıssé limit $M_{\mathrm{F}}$. Here we show that the theory $T_{\mathrm{Sq}}^{\ast }$ of $M_{\mathrm{F}}$ is the model completion of the theory of Steiner triple systems. We also prove that $T_{\mathrm{Sq}}^{\ast }$ is not small and it has quantifier elimination, ${\mathrm{TP}}_2$, ${\mathrm{NSOP}}_1$, elimination of hyperimaginaries and weak elimination of imaginaries.

Cited by in corpus (3)

• Forking, Imaginaries and other features of ACFG
• Towards a Finer Classification of Strongly Minimal Sets
• Countable homogeneous Steiner triple systems avoiding specified subsystems