Model theory of Steiner triple systems

Silvia Barbina, Enrique Casanovas

Published 2018-05-17Version 1

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 $T^\ast_\mathrm{Sq}$ has quantifier elimination, it is not small and has $\mathrm{TP}_2$ and $\mathrm{NSOP}_1$.