The barrier Ramsey theorem

TL;DR

This paper introduces the barrier Ramsey theorem, extending finite Ramsey theory by incorporating barrier largeness and computing associated Ramsey ordinals using advanced ordinal analysis.

Contribution

It develops the concept of barrier largeness for homogeneous sets and calculates Ramsey ordinals through iterations of Veblen functions, advancing the understanding of largeness notions in Ramsey theory.

Findings

Defined barrier largeness as a new notion for homogeneous sets.

Computed Ramsey ordinals using Veblen function iterations.

Extended finite Ramsey theorems to more general largeness conditions.

Abstract

In this paper we study a very general finite Ramsey theorem, where both the sets being colored and the homogeneous set must satisfy some largeness notion. For the homogeneous set this has already been done using the notion of $\alpha$-largeness, where $\alpha$ is a countable ordinal equipped with a system of fundamental sequences. To extend this approach the more appropriate notion is barrier largeness. Since the complexity of barriers can be measured by countable ordinals, we define Ramsey ordinals and, using appropriate iterations of the Veblen functions, we are able to compute them.