Abstract approach to Ramsey theory and Ramsey theorems for finite trees

TL;DR

This paper presents an abstract framework for finite Ramsey theory, unifying and generalizing existing theorems for regular and boron trees, and illustrating its application to derive the Milliken Ramsey theorem.

Contribution

It introduces a new generalization that unifies Deuber's and Jasinski's Ramsey theorems for finite trees, expanding the theoretical landscape.

Findings

Unified framework for tree Ramsey theorems

New generalization of existing theorems

Derivation of Milliken's theorem from the framework

Abstract

I will give a presentation of an abstract approach to finite Ramsey theory found in an earlier paper of mine. I will prove from it a common generalization of Deuber's Ramsey theorem for regular trees and a recent Ramsey theorem of Jasinski for boron tree structures. This generalization appears to be new. I will also show, in exercises, how to deduce from it the Milliken Ramsey theorem for strong subtrees.