Countable partitions of the sets of points and lines

• Autorzy: James H. Schmerl
• Języki publikacji: EN

Abstrakty

EN

The following theorem is proved, answering a question raised by Davies in 1963. If $L_0 ∪ L_1 ∪ L_2 ∪...$ is a partition of the set of lines of $ℝ^n$, then there is a partition $ℝ^n = S_0 ∪ S_1 ∪ S_2 ∪...$ such that $|ℓ ∩ S_i| ≤ 2$ whenever $ℓ ∈ L_i$. There are generalizations to some other, higher-dimensional subspaces, improving recent results of Erdős, Jackson & Mauldin.

Słowa kluczowe

EN

infinite partitions; Euclidean space

Kategorie tematyczne

• 03E05: Other combinatorial set theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1999
• Tom: 160
• Numer: 2
• Strony: 183-196

Daty

wydano

1999

poprawiono

1998-12--02

Twórcy

autor

James H. Schmerl

• Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269, U.S.A.

Bibliografia

• [1] R. O. Davies, On a problem of Erdős concerning decompositions of the plane, Proc. Cambridge Philos. Soc. 59 (1963), 33-36.
• [2] R. O. Davies, On a denumerable partition problem of Erdős, ibid., 501-502.
• [3] P. Erdős, Some remarks on set theory IV, Michigan Math. J. 2 (1953-54), 169-173.
• [4] P. Erdős, S. Jackson and R. D. Mauldin, On partitions of lines and space, Fund. Math. 145 (1994), 101-119.
• [5] P. Erdős, S. Jackson and R. D. Mauldin, On infinite partitions of lines and space, ibid. 152 (1997), 75-95.
• [6] J. H. Schmerl, Countable partitions of Euclidean space, Math. Proc. Cambridge Philos. Soc. 120 (1996), 7-12.
• [7] W. Sierpiński, Sur un théorème équivalent à l'hypothèse du continu $(2^ℵ_0 = ℵ_1)$, Bull. Internat. Acad. Polon. Sci. Lett. Cl. Sci. Math. Nat. Sér. A Sci. Math. 1919, 1-3.
• [8] J. C. Simms, Sierpiński's Theorem, Simon Stevin 65 (1991), 69-163.