On a topological Ramsey Theorem

TL;DR

This paper introduces the $r$-Ramsey property as a strengthening of sequential compactness, proves that metrizable compact spaces satisfy this property for all $r$, and provides examples of spaces that are $r$-Ramsey but not $(r+1)$-Ramsey under CH.

Contribution

It defines the $r$-Ramsey property, establishes its validity for metrizable compact spaces, and constructs examples distinguishing different levels of the property.

Findings

Metrizable compact spaces are $r$-Ramsey for all $r$.

Existence of compact spaces that are $r$-Ramsey but not $(r+1)$-Ramsey under CH.

Introduction of a hierarchy of topological properties related to sequential compactness.

Abstract

We introduce natural strengthenings of sequential compactness called the $r$-Ramsey property for each natural number $r\geq 1$. We prove that metrizable compact spaces are $r$-Ramsey for all $r$ and give examples of compact spaces that are $r$-Ramsey but not $r+1$-Ramsey for each $r\geq 1$ (assuming CH for all $r>1$