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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 182.33101

**Autor: ** Erdös, Pál

**Title: ** Geometrical and set-theoretical properties of subsets of Hilbert-space (In Hungarian)

**Source: ** Mat. Lapok 19, 255-257 (1968).

**Review: ** The author proved in a previous paper without assuming the continuum-hypothesis that if S is a subset of an n-dimensional space then S contains a subset S_{1} of power m so that all the distances between points of S_{1} are distinct. C × t o-by, Kakatuani and the author showed that if P is any denumerable dense set of positive numbers then there is a set H in a Hilbert space of power \aleph_{1} so that all the distances between points in H are in P, further there is a set H_{1} of power C in Hilbert space so that all the distances between points in H_{1} are the square root of a rational number. We do not know if all the distances can be rational.

Is it true that if H is a set of power m in a Hilbert space then H has a subset of power m which does not contain an equilateral triangle?

**Classif.: ** * 46C05 Geometry and topology of inner product spaces

**Index Words: ** set theory

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