# Spreading linear triple systems and expander triple systems

## TL;DR

This paper explores the properties of Steiner triple systems, introducing expander and spreading properties in hypergraphs, and determines minimal sizes for systems with these features, linking to connectivity and influence maximization.

## Contribution

It generalizes known results by defining new properties in hypergraphs and establishing minimal sizes for systems exhibiting these properties, with implications for finite geometry and influence maximization.

## Key findings

- Existence of Steiner triple systems as almost perfect expanders.
- Determined minimal size of linear triple systems with spreading properties.
- Connected to influence maximization and finite geometry applications.

## Abstract

The existence of Steiner triple systems STS(n) of order n containing no nontrivial subsystem is well known for every admissible n. We generalize this result in two ways. First we define the expander property of 3-uniform hypergraphs and show the existence of Steiner triple systems which are almost perfect expanders. Next we define the strong and weak spreading property of linear hypergraphs, and determine the minimum size of a linear triple system with these properties, up to a small constant factor. This property is strongly connected to the connectivity of the structure and of the so-called influence maximization. We also discuss how the results are related to Erd\H{o}s' conjecture on locally sparse STSs, influence maximization, subsquare-free Latin squares and possible applications in finite geometry.