Countable partitions of the sets of points and lines

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Title

Authors ¹

Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269, U.S.A.

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

| ℓ \cap S_{i} | \leq 2

whenever

ℓ \in L_{i}

. There are generalizations to some other, higher-dimensional subspaces, improving recent results of Erdős, Jackson & Mauldin.

infinite partitions, Euclidean space

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