All These Approximate Ramsey Properties

TL;DR

This paper investigates approximate versions of the Ramsey property in finite structures, demonstrating a hierarchy among these properties and characterizing classes where small structures are Ramsey objects, extending known results.

Contribution

It introduces a hierarchy of finitary approximate Ramsey properties and characterizes classes with small Ramsey objects as ordered structures, generalizing prior findings.

Findings

Finitary approximate Ramsey properties form a strict hierarchy.

Classes with all structures of size ≤2 as Ramsey objects are essentially ordered.

Established a connection between small structures and ordered classes in Ramsey theory.

Abstract

We consider finitary approximations of the (embedding) Ramsey property. Using a class of homogeneous reducts of random ordered hypergraphs, we prove that these properties form a strict hierarchy. We also show that every class of finite structures in which every structure of size at most 2 is a "Ramsey object" essentially consists of ordered structures, generalising a known result for countable Ramsey classes.