Some results connected with a problem of Erdős. II

Author:
Harry I. Miller

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

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Abstract: It is shown, using the continuum hypothesis, that if *E* is an uncountable subset of the real line, then there exist subsets and of the unit interval, such that has outer Lebesgue measure one and is of the second Baire category and such that neither nor contains a subset similar (in the sense of elementary geometry) to *E*. These results are related to a conjecture of P. Erdős.

**[1]**Alexander Abian,*Partition of nondenumerable closed sets of reals*, Czechoslovak Math. J.**26(101)**(1976), no. 2, 207–210. MR**0401997****[2]**P. Erdős,*Problems*, Math. Balkanica (Papers presented at The Fifth Balkan Mathematical Congress),**4**(1974), 203-204.**[3]**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****[4]**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).**[5]**M. S. Ruziewicz,*Contribution à l'étude des ensembles de distances de points*, Fund. Math.**7**(1925), 141-143.**[6]**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.

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DOI:
https://doi.org/10.1090/S0002-9939-1979-0532148-4

Article copyright:
© Copyright 1979
American Mathematical Society