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**Zentralblatt MATH**

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

**Zbl.No: ** 418.04002

**Autor: ** Erdös, Paul

**Title: ** Set theoretic, measure theoretic, combinatorial, and number theoretic problems concerning point sets in Euclidean space. (In English)

**Source: ** Real Anal. Exch. 4(1978-79), 113-138 (1979).

**Review: ** The author states several combinatorial statements putting in contrast the corresponding situations for the finite and the infinite case respectively and that in general the case of finite sets is more complicate than the one of transfinite sets. E.g. if S is any infinite set in E_{k}, then S contains an equinumerous set S_{1} such that all distances between points of S_{1} are distinct (cf. *P.Erdös*, Proc. Am. Math. Soc. 1, 127-141 (1950; Zbl 039.04902). On the other hand, let f_{k}(n) be the largest integer such that any n-point-set S in E_{k} contains an f_{k}(n)-subset S_{1} such that all distances of points of S_{1} are distinct. ''The exact determination of f_{k}(n) seems hopoless...''. A plausible conjecture is f_{1}(n) = (1+o(1))n^{ ½}, where g_{1}(n) = **max** k such that any strictly increasing k-sequence a_{1} < a_{2} < ... < a_{k} of natural numbers \leq n has all distinct differences a_{j}-a_{i}. The author conjectures that g_{1}(n) = n^{ ½}+0(1). He conjectures that f_{1}(n) = g_{1}(n) = n^{ ½}+0(1). Let n_{k} be the smallest integer such that f_{k}(n_{k}) = 3; e.g. n_{2} = 9; it is not known whether n_{k}^{1/k} ––> 1. Problem: Is there a constant C > 0 such that every set s\subset E_{2} of measure > C contains the vertices of a triangle area? Problem: Given a countable subset A of [0,1]; estimate the largest possible measure of a subset of [0,1] which does not contain a set similar to A.

Problem: A set S in an Euclidean space of finite dimension is said to be Ramsey if for every k in N there is an n_{k} such that if E_{nk} is decomposed into k disjoint sets A_{i} then S is contained in one of these sets A_{1},..., A_{k}; is every obtuse angled triangle Ramsey? Is regular pentagon Ramsey? Problem: Is there a set S of power c in Hilbert space such that every equinumerous subset S_{1} contains an equilateral triangle (resp. an infinite dimensional regular simplex)? Many other questions are discoussed in the present paper announcing that he matter will be extensively discussed along with other questions in a forthcoming book written jointly by the author and George Purdy.

**Reviewer: ** \D.Kurepa

**Classif.: ** * 04A05 Relations, functions

04A10 Ordinal and cardinal numbers; generalizations

04A20 Combinatorial set theory

05A17 Partitions of integres (combinatorics)

28A99 Classical measure theory

00A07 Problem books

**Keywords: ** combinatorial problems in set theory; different distances; subsets of Euclidean space; partitions; Hilbert space; measures

**Citations: ** Zbl.039.049

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