The search session has expired. Please query the service again.

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

top
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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.