On partitions of lines and space

Paul Erdös; Steve Jackson; R. Mauldin

Fundamenta Mathematicae (1994)

  • Volume: 145, Issue: 2, page 101-119
  • ISSN: 0016-2736

Abstract

top
We consider a set, L, of lines in ℝ n and a partition of L into some number of sets: L = L 1 ∪ . . . ∪ L p . We seek a corresponding partition ℝ n = S 1 ∪ . . . ∪ S p such that each line l in L i meets the set S i in a set whose cardinality has some fixed bound, ω τ . We determine equivalences between the bounds on the size of the continuum, 2 ω ≤ ω θ , and some relationships between p, ω τ and ω θ .

How to cite

top

Erdös, Paul, Jackson, Steve, and Mauldin, R.. "On partitions of lines and space." Fundamenta Mathematicae 145.2 (1994): 101-119. <http://eudml.org/doc/212037>.

@article{Erdös1994,
abstract = {We consider a set, L, of lines in $ℝ^n$ and a partition of L into some number of sets: $L = L_1∪...∪ L_p$. We seek a corresponding partition $ℝ^n = S_1 ∪...∪ S_p$ such that each line l in $L_i$ meets the set $S_i$ in a set whose cardinality has some fixed bound, $ω_τ$. We determine equivalences between the bounds on the size of the continuum, $2^ω ≤ ω_θ$, and some relationships between p, $ω_τ$ and $ω_θ$.},
author = {Erdös, Paul, Jackson, Steve, Mauldin, R.},
journal = {Fundamenta Mathematicae},
keywords = {transfinite recursion; value of continuum; partitions of lines in Euclidean space; axiom of determinateness},
language = {eng},
number = {2},
pages = {101-119},
title = {On partitions of lines and space},
url = {http://eudml.org/doc/212037},
volume = {145},
year = {1994},
}

TY - JOUR
AU - Erdös, Paul
AU - Jackson, Steve
AU - Mauldin, R.
TI - On partitions of lines and space
JO - Fundamenta Mathematicae
PY - 1994
VL - 145
IS - 2
SP - 101
EP - 119
AB - We consider a set, L, of lines in $ℝ^n$ and a partition of L into some number of sets: $L = L_1∪...∪ L_p$. We seek a corresponding partition $ℝ^n = S_1 ∪...∪ S_p$ such that each line l in $L_i$ meets the set $S_i$ in a set whose cardinality has some fixed bound, $ω_τ$. We determine equivalences between the bounds on the size of the continuum, $2^ω ≤ ω_θ$, and some relationships between p, $ω_τ$ and $ω_θ$.
LA - eng
KW - transfinite recursion; value of continuum; partitions of lines in Euclidean space; axiom of determinateness
UR - http://eudml.org/doc/212037
ER -

References

top
  1. [B] F. Bagemihl, A proposition of elementary plane geometry that implies the continuum hypothesis, Z. Math. Logik Grundlag. Math. 7 (1961), 77-79. Zbl0121.01604
  2. [BH] G. Bergman and E. Hrushovski, Identities of cofinal sublattices, Order 2 (1985), 173-191. Zbl0579.06010
  3. [D1] R. Davies, On a problem of Erdős concerning decompositions of the plane, Proc. Cambridge Philos. Soc. 59 (1963), 33-36. Zbl0121.25702
  4. [D2] R. Davies, On a denumerable partition problem of Erdős, ibid., 501-502. Zbl0121.01602
  5. [Er] P. Erdős, Some remarks on set theory. IV, Michigan Math. J. 2 (1953-54), 169-173. 
  6. [EGH] P. Erdős, F. Galvin and A. Hajnal, On set-systems having large chromatic number and not containing prescribed subsystems, in: Infinite and Finite Sets (Colloq., Keszthely; dedicated to P. Erdős on his 60th birthday, 1973), Colloq. Math. Soc. János Bolyai 10, North-Holland, Amsterdam, 1975, 425-513. 
  7. [EH] P. Erdős and A. Hajnal, On chromatic number of graphs and set-systems, Acta. Math. Acad. Sci. Hungar. 17 (1966), 61-99. Zbl0151.33701
  8. [GG] F. Galvin and G. Gruenhage, A geometric equivalent of the continuum hypothesis, unpublished manuscript. 
  9. [J1] S. Jackson, A new proof of the strong partition relation on ω 1 , Trans. Amer. Math. Soc. 320 (1990), 737-745. 
  10. [J2] S. Jackson, A computation of δ 5 1 , to appear. 
  11. [Ku] C. Kuratowski, Sur une caractérisation des alephs, Fund. Math. 38 (1951), 14-17. Zbl0044.27302
  12. [S1] W. Sierpiński, Sur quelques propositions concernant la puissance du continu, ibid., 1-13. Zbl0044.27301
  13. [S2] W. Sierpiński, Hypothèse du continu, Warszawa, 1934. Zbl60.0035.01
  14. [Si] R. Sikorski, A characterization of alephs, Fund. Math. 38 (1951), 18-22. Zbl0044.27303
  15. [Sm1] J. Simms, Sierpiński's theorem, Simon Stevin 65 (1991), 69-163. Zbl0726.01013
  16. [Sm2] J. Simms, Is 2 ω weakly inaccessible?, to appear. 
  17. [To] S. Todorčević, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), 261-294. Zbl0658.03028
  18. [W] W. H. Woodin, Some consistency results in ZFC using AD, in: Cabal Seminar 79-81, Lecture Notes in Math. 1019, Springer, 1983, 172-198. 

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.