Some results connected with a problem of Erdős. II
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- by Harry I. Miller PDF
- Proc. Amer. Math. Soc. 75 (1979), 265-268 Request permission
Abstract:
It is shown, using the continuum hypothesis, that if E is an uncountable subset of the real line, then there exist subsets ${S_1}$ and ${S_2}$ of the unit interval, such that ${S_1}$ has outer Lebesgue measure one and ${S_2}$ is of the second Baire category and such that neither ${S_1}$ nor ${S_2}$ contains a subset similar (in the sense of elementary geometry) to E. These results are related to a conjecture of P. Erdős.References
- Alexander Abian, Partition of nondenumerable closed sets of reals, Czechoslovak Math. J. 26(101) (1976), no. 2, 207–210. MR 401997 P. Erdős, Problems, Math. Balkanica (Papers presented at The Fifth Balkan Mathematical Congress), 4 (1974), 203-204.
- Harry I. Miller, Relationships between various gauges of the size of sets of real numbers. II, Akad. Nauka Umjet. Bosne Hercegov. Rad. Odjelj. Prirod. Mat. Nauka 16 (1976), 37–48 (English, with Serbo-Croatian summary). MR 536603 H. I. Miller and P. I. Xenikakis, Some results connected with a problem of Erdős. I, Akad. Nauka i Umjet. Bosne i Hercegov. Rad. Odjelj. Prirod. Mat. Nauka (to appear). M. S. Ruziewicz, Contribution à l’étude des ensembles de distances de points, Fund. Math. 7 (1925), 141-143. W. Sierpiński, Un théorème de la théorie générale des ensembles et ses applications, C. R. Soc. Sci. Varsovie 28 (1936), 131-135. Zbl. 15, 103.
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 75 (1979), 265-268
- MSC: Primary 28A05; Secondary 26A21
- DOI: https://doi.org/10.1090/S0002-9939-1979-0532148-4
- MathSciNet review: 532148