On splitting infinite-fold covers

Tom 212 / 2011

Márton Elekes, Tamás Mátrai, Lajos Soukup Fundamenta Mathematicae 212 (2011), 95-127 MSC: Primary 03E05, 03E15; Secondary 03C25, 03E04, 03E35, 03E40, 03E50, 03E65, 05C15, 06A05, 52A20, 52B11. DOI: 10.4064/fm212-2-1

Streszczenie

Let $X$ be a set, $\kappa$ be a cardinal number and let ${\cal H}$ be a family of subsets of $X$ which covers each $x\in X$ at least $\kappa$-fold. What assumptions can ensure that ${\cal H}$ can be decomposed into $\kappa$ many disjoint subcovers?

We examine this problem under various assumptions on the set $X$ and on the cover ${\cal H}$: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of ${\mathbb R}^{n}$ by polyhedra and by arbitrary convex sets. We focus on problems with $\kappa$ infinite. Besides numerous positive and negative results, many questions turn out to be independent of the usual axioms of set theory.

Autorzy

  • Márton ElekesAlfréd Rényi Institute of Mathematics
    Hungarian Academy of Sciences
    P.O. Box 127, H-1364 Budapest, Hungary
    e-mail
    e-mail
  • Tamás MátraiAlfréd Rényi Institute of Mathematics
    Hungarian Academy of Sciences
    P.O. Box 127, H-1364 Budapest, Hungary
    e-mail
    e-mail
  • Lajos SoukupAlfréd Rényi Institute of Mathematics
    Hungarian Academy of Sciences
    P.O. Box 127, H-1364 Budapest, Hungary
    e-mail
    e-mail

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