On infinite partitions of lines and space

P. Erdos, Steve Jackson, R. Daniel Mauldin

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

Given a partition P : L → ω of the lines in ℝn, n ≥ 2, into countably many pieces, we ask if it is possible to find a partition of the points, Q : ℝn → 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 languageEnglish
Pages (from-to)75-95
Number of pages21
JournalFundamenta Mathematicae
Volume152
Issue number1
Publication statusPublished - Dec 1 1997

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Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Erdos, P., Jackson, S., & Mauldin, R. D. (1997). On infinite partitions of lines and space. Fundamenta Mathematicae, 152(1), 75-95.