Countable homogeneous Steiner triple systems avoiding specified subsystems

TL;DR

This paper constructs numerous new homogeneous infinite Steiner triple systems by employing Fra"{}sse9 limits, avoiding certain subsystems, and introduces a novel embedding result for finite partial systems.

Contribution

It introduces a new embedding theorem for finite partial Steiner systems and constructs uncountably many homogeneous infinite systems avoiding specific subsystems.

Findings

Constructed uncountably many new homogeneous Steiner systems.

Developed a new embedding result for finite partial systems.

Connected the constructions to model-theoretic classification.

Abstract

In this article we construct uncountably many new homogeneous locally finite Steiner triple systems of countably infinite order as Fra\"{\i}ss\'{e} limits of classes of finite Steiner triple systems avoiding certain subsystems. The construction relies on a new embedding result: any finite partial Steiner triple system has an embedding into a finite Steiner triple system that contains no nontrivial proper subsystems that are not subsystems of the original partial system. Fra\"{\i}ss\'e's construction and its variants are rich sources of examples that are central to model-theoretic classification theory, and recently infinite Steiner systems obtained via Fra\"{\i}ss\'e-type constructions have received attention from the model theory community.