Finite big Ramsey degrees in universal structures

TL;DR

This paper explores finite big Ramsey degrees in structures beyond Fra"{i}ss"e limits, demonstrating their finiteness in certain non-Fra"{i}ss"e classes like acyclic oriented graphs using categorical methods.

Contribution

It extends the concept of big Ramsey degrees to non-Fra"{i}ss"e structures, showing their finiteness in classes such as acyclic oriented graphs through categorical techniques.

Findings

Finite big Ramsey degrees exist in non-Fra"{i}ss"e structures.

Acyclic oriented graphs have finite big Ramsey degrees.

Categorical methods are effective in studying Ramsey properties.

Abstract

Big Ramsey degrees of finite structures are usually considered with respect to a Fra\"{i} ss\'e limit. Building mainly on the work of Devlin, Sauer, Laflamme and Van Th\'e, in this paper we consider structures which are not Fra\"{i} ss\'e limits, and still have the property that their finite substructures have finite big Ramsey degrees in them. For example, the class of all finite acyclic oriented graphs is not a Fra\"{i} ss\'e class, and yet we show that there is a countably infinite acyclic oriented graph in which every finite acyclic oriented graph has finite big Ramsey degree. Our main tools come from category theory as it has recently become evident that the Ramsey property is not only a deep combinatorial property, but also a genuine categorical property.