On infinite partitions of lines and space

• Autorzy: Paul Erdös , Steve Jackson , R. Daniel Mauldin
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

transfinite recursion; Martin's Axiom; forcing; geometry; infinite partitions

Kategorie tematyczne

• 03E50: Continuum hypothesis and Martin's axiom
• 03E05: Other combinatorial set theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1997
• Tom: 152
• Numer: 1
• Strony: 75-95

Daty

wydano

1997

otrzymano

1996-06-27

Twórcy

autor

Paul Erdös

• Mathematical Institute, Hungarian Academy of Sciences, Reáltanoda U. 13-15, H-1053 Budapest, Hungary

autor

Steve Jackson

• Department of Mathematics, University of North Texas, Denton, Texas 76203-5116, U.S.A.,

autor

R. Daniel Mauldin

• Department of Mathematics, University of North Texas, Denton, Texas 76203-5116, U.S.A.,

Bibliografia

• [1] R. Davies, On a denumerable partition problem of Erdős, Proc. Cambridge Philos. Soc. 59 (1963), 33-36.
• [2] P. Erdős, S. Jackson and R. D. Mauldin, On partitions of lines and space, Fund. Math. 145 (1994), 101-119.
• [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.
• [6] S. Todorčević, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), 261-294.