Erd\H{o}s-Rado Classes

TL;DR

This paper introduces the concept of combinatorial Erd ext{"o}s-Rado classes, merging Ramsey classes and the Erd ext{"o}s-Rado Theorem, with applications to model theory and non-elementary classes.

Contribution

It defines combinatorial Erd ext{"o}s-Rado classes and explores their properties and applications in constructing indiscernibles in Abstract Elementary Classes.

Findings

Introduction of combinatorial Erd ext{"o}s-Rado classes

Examples of such classes provided

Applications to model theory and indiscernibles

Abstract

We amalgamate two generalizations of Ramsey's Theorem--Ramsey classes and the Erd\H{o}s-Rado Theorem--into the notion of a combinatorial Erd\H{o}s-Rado class. These classes are closely related to Erd\H{o}s-Rado classes, which are those from which we can build generalized indiscernibles and blueprints in nonelementary classes, especially Abstract Elementary Classes. We give several examples and some applications.