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A+ CATEGORY SCIENTIFIC UNIT

Coloring grids

Volume 228 / 2015

Ramiro de la Vega Fundamenta Mathematicae 228 (2015), 283-289 MSC: Primary 03E50; Secondary 03E05, 51M05. DOI: 10.4064/fm228-3-5

Abstract

A structure where each E_i is an equivalence relation on A is called an n-grid if any two equivalence classes coming from distinct E_i's intersect in a finite set. A function \chi : A \to n is an acceptable coloring if for all i \in n, the \chi ^{-1}(i) intersects each E_i-equivalence class in a finite set. If B is a set, then the n-cube B^n may be seen as an n-grid, where the equivalence classes of E_i are the lines parallel to the ith coordinate axis. We use elementary submodels of the universe to characterize those n-grids which admit an acceptable coloring. As an application we show that if an n-grid \mathcal {A} does not admit an acceptable coloring, then every finite n-cube is embeddable in \mathcal {A}.

Authors

  • Ramiro de la VegaDepartamento de Matemáticas
    Universidad de los Andes
    Cra 1 No. 18A-12
    Bogotá, Colombia
    e-mail

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