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The following theorem is proved, answering a question raised by Davies in 1963. If L 0 ∪ L 1 ∪ L 2 ∪ . . . is a partition of the set of lines of R n , then there is a partition R n = S 0 ∪ S 1 ∪ S 2 ∪ . . . such that | ∩ S i | ≤ 2 whenever ∈ L i . There are generalizations to some other, higher-dimensional subspaces, improving recent results of Erdős, Jackson & Mauldin.doi:10.4064/fm-160-2-183-196 fatcat:ha3xeh6h5nbinmt7333owgsaky