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2 Fundamenta Mathematicae

2 de la Vega R.

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Coloring grids

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de la Vega R.

Fundamenta Mathematicae

2015

tom 228

nr 3

283-289

A structure $𝓐 = (A;E_{i})_{i∈n}$ 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 χ: A → n is an acceptable coloring if for all i ∈ n, the $χ^{-1}(i)$ intersects each $E_{i}$-equivalence class in a finite set. If B is a set, then the n-cube Bⁿ 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 𝓐 does not admit an acceptable coloring, then every finite n-cube is embeddable in 𝓐.

Decompositions of the plane and the size of the continuum

100%

de la Vega R.

Fundamenta Mathematicae

2009

tom 203

nr 1

65-74

We consider a triple ⟨E₀,E₁,E₂⟩ of equivalence relations on ℝ² and investigate the possibility of decomposing the plane into three sets ℝ² = S₀ ∪ S₁ ∪ S₂ in such a way that each $S_i$ intersects each $E_i$-class in finitely many points. Many results in the literature, starting with a famous theorem of Sierpiński, show that for certain triples the existence of such a decomposition is equivalent to the continuum hypothesis. We give a characterization in ZFC of the triples for which the decomposition exists. As an application we show that the plane can be covered by three sprays regardless of the size of the continuum, thus answering a question of J. H. Schmerl.