The following theorem is proved, answering a question raised by Davies in 1963. If $L_0\cup L_1\cup L_2\cup ...$ is a partition of the set of lines of ${ℝ}^n$, then there is a partition ${ℝ}^n=S_0\cup S_1\cup S_2\cup ...$ such that $|\ell \cap S_i|\le 2$ whenever $\ell \in L_i$. There are generalizations to some other, higher-dimensional subspaces, improving recent results of Erdős, Jackson Mauldin.

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.