A structure where each is an equivalence relation on A is called an n-grid if any two equivalence classes coming from distinct ’s intersect in a finite set. A function χ: A → n is an acceptable coloring if for all i ∈ n, the intersects each -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 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 intersects each -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....
Download Results (CSV)