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 Ek, then S contains an equinumerous set S1 such that all distances between points of S1 are distinct (cf. P.Erdös, Proc. Am. Math. Soc. 1, 127-141 (1950; Zbl 039.04902). On the other hand, let fk(n) be the largest integer such that any n-point-set S in Ek contains an fk(n)-subset S1 such that all distances of points of S1 are distinct. ''The exact determination of fk(n) seems hopoless...''. A plausible conjecture is f1(n) = (1+o(1))n ½, where g1(n) = max k such that any strictly increasing k-sequence a1 < a2 < ... < ak of natural numbers \leq n has all distinct differences aj-ai. The author conjectures that g1(n) = n ½+0(1). He conjectures that f1(n) = g1(n) = n ½+0(1). Let nk be the smallest integer such that fk(nk) = 3; e.g. n2 = 9; it is not known whether nk1/k ––> 1. Problem: Is there a constant C > 0 such that every set s\subset E2 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 nk such that if Enk is decomposed into k disjoint sets Ai then S is contained in one of these sets A1,..., Ak; 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 S1 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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