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A structure $={(A;E_i)}_{i\in n}$ 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 ${\chi }^{-1}\left(i\right)$ 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...

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....