Ramsey theory and the borel conjecture

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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 σ-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 σ-compact is explored.

Original languageEnglish
Pages (from-to)69-113
Number of pages45
JournalContemporary Mathematics
Volume755
DOIs
StatePublished - 2020

Keywords

  • Infinite game
  • Ramsey theory
  • Rothberger bounded
  • Strong measure zero
  • Topological group
  • Uni-formizable space

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