### Abstract

Given a partition P : L → ω of the lines in ℝ^{n}, n ≥ 2, into countably many pieces, we ask if it is possible to find a partition of the points, Q : ℝ^{n} → ω, so that each line meets at most m points of its color. Assuming Martin's Axiom, we show this is the case for m ≥ 3. We reduce the problem for m = 2 to a purely finitary geometry problem. Although we have established a very similar, but somewhat simpler, version of the geometry conjecture, we leave the general problem open. We consider also various generalizations of these results, including to higher dimension spaces and planes.

Original language | English |
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Pages (from-to) | 75-95 |

Number of pages | 21 |

Journal | Fundamenta Mathematicae |

Volume | 152 |

Issue number | 1 |

Publication status | Published - Dec 1 1997 |

### Keywords

- Forcing
- Geometry
- Infinite partitions
- Martin's Axiom
- Transfinite recursion

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Erdos, P., Jackson, S., & Mauldin, R. D. (1997). On infinite partitions of lines and space.

*Fundamenta Mathematicae*,*152*(1), 75-95.