Ramsey Theory and the Borel Conjecture

TL;DR

This paper explores the deep connections between the Borel covering property and Ramsey theory, analyzing their interplay in various types of topological spaces and introducing stronger covering properties.

Contribution

It establishes new links between Borel covering properties and Ramsey theory, including for stronger properties like Rothberger's, and characterizes these in $\sigma$-compact spaces.

Findings

Connection between Borel covering property and Ramsey theory in $\sigma$-compact spaces

Landmark Ramseyan theorems characterize stronger covering properties

Analysis of the case when the space with the stronger property is $\sigma$-compact

Abstract

The Borel covering property, introduced a century ago by E. Borel, is intimately connected with Ramsey theory, initiated ninety years ago in an influential paper of F.P. Ramsey. The current state of knowledge about the connection between the Borel covering property and Ramsey theory is outlined in this paper. Initially the connection is established for the situation when the set with the Borel covering property is a proper subset of a $\sigma$-compact uniform space. Then the connection is explored for a stronger covering property introduced by Rothberger. After establishing the fact that in this case several landmark Ramseyan theorems are characteristic of this stronger covering property, the case when the space with this stronger covering property is in fact $\sigma$-compact is explored.