# 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

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topErdös, 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

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

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