On infinite partitions of lines and space

Paul Erdös; Steve Jackson; R. Mauldin

Fundamenta Mathematicae (1997)

  • Volume: 152, Issue: 1, page 75-95
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Erdös, Paul, Jackson, Steve, and Mauldin, R.. "On infinite partitions of lines and space." Fundamenta Mathematicae 152.1 (1997): 75-95. <http://eudml.org/doc/212200>.

@article{Erdös1997,
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.},
author = {Erdös, Paul, Jackson, Steve, Mauldin, R.},
journal = {Fundamenta Mathematicae},
keywords = {transfinite recursion; Martin's Axiom; forcing; geometry; infinite partitions; Martin's axiom; set theoretic constructions in euclidean spaces},
language = {eng},
number = {1},
pages = {75-95},
title = {On infinite partitions of lines and space},
url = {http://eudml.org/doc/212200},
volume = {152},
year = {1997},
}

TY - JOUR
AU - Erdös, Paul
AU - Jackson, Steve
AU - Mauldin, R.
TI - On infinite partitions of lines and space
JO - Fundamenta Mathematicae
PY - 1997
VL - 152
IS - 1
SP - 75
EP - 95
AB - 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.
LA - eng
KW - transfinite recursion; Martin's Axiom; forcing; geometry; infinite partitions; Martin's axiom; set theoretic constructions in euclidean spaces
UR - http://eudml.org/doc/212200
ER -

References

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  1. [1] R. Davies, On a denumerable partition problem of Erdős, Proc. Cambridge Philos. Soc. 59 (1963), 33-36. Zbl0121.25702
  2. [2] P. Erdős, S. Jackson and R. D. Mauldin, On partitions of lines and space, Fund. Math. 145 (1994), 101-119. Zbl0809.04004
  3. [3] S. Jackson and R. D. Mauldin, Set Theory and Geometry, to appear. 
  4. [4] T. Jech, Set Theory, Academic Press, 1978. 
  5. [5] K. Kunen, Set Theory, an Introduction to Independence Proofs, North-Holland, 1980. Zbl0443.03021
  6. [6] S. Todorčević, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), 261-294. Zbl0658.03028

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