For any three noncollinear points c₀,c₁,c₂ ∈ ℝ², there are sprays S₀,S₁,S₂ centered at c₀,c₁,c₂ that cover ℝ². This improves the result of de la Vega in which c₀,c₁,c₂ were required to be the vertices of an equilateral triangle.
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....