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A structure $𝓐 = (A;E_{i})_{i∈n}$ where each 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...
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 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...
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