# On partitions of lines and space

Fundamenta Mathematicae (1994)

• Volume: 145, Issue: 2, page 101-119
• ISSN: 0016-2736

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## Abstract

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We consider a set, L, of lines in ${ℝ}^{n}$ and a partition of L into some number of sets: $L={L}_{1}\cup ...\cup {L}_{p}$. We seek a corresponding partition ${ℝ}^{n}={S}_{1}\cup ...\cup {S}_{p}$ such that each line l in ${L}_{i}$ meets the set ${S}_{i}$ in a set whose cardinality has some fixed bound, ${\omega }_{\tau }$. We determine equivalences between the bounds on the size of the continuum, ${2}^{\omega }\le {\omega }_{\theta }$, and some relationships between p, ${\omega }_{\tau }$ and ${\omega }_{\theta }$.

## How to cite

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Erdös, Paul, Jackson, Steve, and Mauldin, R.. "On partitions of lines and space." Fundamenta Mathematicae 145.2 (1994): 101-119. <http://eudml.org/doc/212037>.

@article{Erdös1994,
abstract = {We consider a set, L, of lines in $ℝ^n$ and a partition of L into some number of sets: $L = L_1∪...∪ L_p$. We seek a corresponding partition $ℝ^n = S_1 ∪...∪ S_p$ such that each line l in $L_i$ meets the set $S_i$ in a set whose cardinality has some fixed bound, $ω_τ$. We determine equivalences between the bounds on the size of the continuum, $2^ω ≤ ω_θ$, and some relationships between p, $ω_τ$ and $ω_θ$.},
author = {Erdös, Paul, Jackson, Steve, Mauldin, R.},
journal = {Fundamenta Mathematicae},
keywords = {transfinite recursion; value of continuum; partitions of lines in Euclidean space; axiom of determinateness},
language = {eng},
number = {2},
pages = {101-119},
title = {On partitions of lines and space},
url = {http://eudml.org/doc/212037},
volume = {145},
year = {1994},
}

TY - JOUR
AU - Erdös, Paul
AU - Jackson, Steve
AU - Mauldin, R.
TI - On partitions of lines and space
JO - Fundamenta Mathematicae
PY - 1994
VL - 145
IS - 2
SP - 101
EP - 119
AB - We consider a set, L, of lines in $ℝ^n$ and a partition of L into some number of sets: $L = L_1∪...∪ L_p$. We seek a corresponding partition $ℝ^n = S_1 ∪...∪ S_p$ such that each line l in $L_i$ meets the set $S_i$ in a set whose cardinality has some fixed bound, $ω_τ$. We determine equivalences between the bounds on the size of the continuum, $2^ω ≤ ω_θ$, and some relationships between p, $ω_τ$ and $ω_θ$.
LA - eng
KW - transfinite recursion; value of continuum; partitions of lines in Euclidean space; axiom of determinateness
UR - http://eudml.org/doc/212037
ER -

## References

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