# Countable partitions of the sets of points and lines

Fundamenta Mathematicae (1999)

• Volume: 160, Issue: 2, page 183-196
• ISSN: 0016-2736

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## Abstract

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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.

## How to cite

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Schmerl, James. "Countable partitions of the sets of points and lines." Fundamenta Mathematicae 160.2 (1999): 183-196. <http://eudml.org/doc/212387>.

@article{Schmerl1999,
abstract = {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.},
author = {Schmerl, James},
journal = {Fundamenta Mathematicae},
keywords = {infinite partitions; Euclidean space; set of lines of ; countable partition; set of points; higher-dimensional subspaces},
language = {eng},
number = {2},
pages = {183-196},
title = {Countable partitions of the sets of points and lines},
url = {http://eudml.org/doc/212387},
volume = {160},
year = {1999},
}

TY - JOUR
AU - Schmerl, James
TI - Countable partitions of the sets of points and lines
JO - Fundamenta Mathematicae
PY - 1999
VL - 160
IS - 2
SP - 183
EP - 196
AB - 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.
LA - eng
KW - infinite partitions; Euclidean space; set of lines of ; countable partition; set of points; higher-dimensional subspaces
UR - http://eudml.org/doc/212387
ER -

## References

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1. [1] R. O. Davies, On a problem of Erdős concerning decompositions of the plane, Proc. Cambridge Philos. Soc. 59 (1963), 33-36. Zbl0121.25702
2. [2] R. O. Davies, On a denumerable partition problem of Erdős, ibid., 501-502. Zbl0121.01602
3. [3] P. Erdős, Some remarks on set theory IV, Michigan Math. J. 2 (1953-54), 169-173.
4. [4] P. Erdős, S. Jackson and R. D. Mauldin, On partitions of lines and space, Fund. Math. 145 (1994), 101-119. Zbl0809.04004
5. [5] P. Erdős, S. Jackson and R. D. Mauldin, On infinite partitions of lines and space, ibid. 152 (1997), 75-95. Zbl0883.03031
6. [6] J. H. Schmerl, Countable partitions of Euclidean space, Math. Proc. Cambridge Philos. Soc. 120 (1996), 7-12. Zbl0887.51013
7. [7] W. Sierpiński, Sur un théorème équivalent à l’hypothèse du continu $\left({2}_{0}^{\aleph }={\aleph }_{1}\right)$, Bull. Internat. Acad. Polon. Sci. Lett. Cl. Sci. Math. Nat. Sér. A Sci. Math. 1919, 1-3.
8. [8] J. C. Simms, Sierpiński's Theorem, Simon Stevin 65 (1991), 69-163. Zbl0726.01013

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