A Ramsey Class for Steiner Systems

TL;DR

This paper constructs a specialized Ramsey class for Steiner systems, introducing the concept of strongly induced subsystems, and explores their Ramsey properties, including an induced Ramsey theorem for designs.

Contribution

It introduces a new Ramsey class for Steiner systems with the notion of strongly induced subsystems, expanding understanding of Ramsey properties in combinatorial design classes.

Findings

Established a Ramsey class for Steiner systems.

Defined and utilized strongly induced subsystems.

Proved an induced Ramsey theorem for designs.

Abstract

We construct a Ramsey class whose objects are Steiner systems. In contrast to the situation with general $r$-uniform hypergraphs, it turns out that simply putting linear orders on their sets of vertices is not enough for this purpose: one also has to strengthen the notion of subobjects used from "induced subsystems" to something we call "strongly induced subsystems". Moreover we study the Ramsey properties of other classes of Steiner systems obtained from this class by either forgetting the order or by working with the usual notion of subsystems. This leads to a perhaps surprising induced Ramsey theorem in which designs get coloured.