Geodesic
Countable partitions of the sets of points and lines
Fundamenta Mathematicae, Tome 160 (1999) no. 2, pp. 183-196

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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 $|ℓ ∩ S_i| ≤ 2$ whenever $ℓ ∈ L_i$. There are generalizations to some other, higher-dimensional subspaces, improving recent results of Erdős, Jackson Mauldin.
DOI : 10.4064/fm-160-2-183-196
Keywords: infinite partitions, Euclidean space

James H. Schmerl 1

1 
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James H. Schmerl. Countable partitions of the sets of points and lines. Fundamenta Mathematicae, Tome 160 (1999) no. 2, pp. 183-196. doi: 10.4064/fm-160-2-183-196

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