Algebra, selections, and additive Ramsey theory

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, can be viewed as coloring theorems concerning countable covers of countable, discrete sets. These theorems are extended to covers of arbitrary topological spaces with Menger's classical covering property. The methods include, in addition to Hurewicz's game-theoretic characterization of Menger's property, extensions of the classical idempotent theory in the Stone-Čech compactification of semigroups, and of the more recent theory of selection principles. This provides strong versions of the mentioned celebrated theorems, where the monochromatic substructures are large, beyond infinitude, in an analytical sense. Reducing the main theorems to the purely combinatorial setting, we obtain nontrivial consequences concerning uncountable cardinal characteristics of the continuum. The main results, modulo technical adjustments, are of the following type (definitions provided in the main text): Let X be a Menger space, and U be an infinite open cover of X. Consider the complete graph whose vertices are the open sets in X. For each finite coloring of the vertices and edges of this graph there are disjoint finite subsets F 1,F2,⋯ of the cover U whose unions V1:= ∪ F1, V2:= ∪ F 2,⋯ have the following properties: (1) The sets ∪nϵF Vn and ∪nϵH Vn are distinct for all nonempty finite sets F < H. (2) All vertices ∪nϵF Vn for nonempty finite sets F have the same color. (3) All edges {∪nϵF Vn, ∪nϵH Vn } for nonempty finite sets F < H have the same color. (4) The family {V1, V2,⋯} forms a cover of X. A self-contained introduction to the necessary parts of the needed theories is included.

Original languageEnglish
Pages (from-to)81-104
Number of pages24
JournalFundamenta Mathematicae
Volume240
Issue number1
DOIs
StatePublished - 2018

Bibliographical note

Publisher Copyright:
© Instytut Matematyczny PAN, 2018.

Keywords

  • Hindman's Finite Sums Theorem
  • Infinite topological games
  • Menger space
  • Milliken-Taylor Theorem
  • Selection principles
  • Stone-Čech compactification

Fingerprint

Dive into the research topics of 'Algebra, selections, and additive Ramsey theory'. Together they form a unique fingerprint.

Cite this