Steiner triple systems and spreading sets in projective spaces

TL;DR

This paper investigates the extremal properties of spreading sets in Steiner triple systems, establishing bounds and characterizing systems with large minimal spreading sets, notably linking them to projective spaces.

Contribution

It provides sharp bounds on minimal spreading set sizes and characterizes systems with large minimal spreading sets as projective spaces, revealing structural insights.

Findings

Sharp upper bounds on minimal spreading set sizes

Large minimal spreading sets imply the system is a projective space

Minimal spreading set size is not an invariant of Steiner triple systems

Abstract

We address several extremal problems concerning the spreading property of point sets of Steiner triple systems. This property is closely related to the structure of subsystems, as a set is spreading if and only if there is no proper subsystem which contains it. We give sharp upper bounds on the size of a minimal spreading set in a Steiner triple system and show that if all the minimal spreading sets are large then the examined triple system must be a projective space. We also show that the size of a minimal spreading set is not an invariant of a Steiner triple system.