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**Zentralblatt MATH**

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

**Zbl.No: ** 575.10041

**Autor: ** Erdös, Paul; Harzheim, E.

**Title: ** Congruent subsets of infinite sets of natural numbers. (In English)

**Source: ** J. Reine Angew. Math. 367, 207-214 (1986).

**Review: ** If A is an infinite subset of the set **N** of natural numbers, A(x) denotes the number of elements of A which are \leq x. The main theorem states: If k and n are given natural numbers > 1 and if A(x) \geq \epsilon · x^{1-1/n} for some positive \epsilon and all x of a final segment of **N**, then there exist k disjoint n-element subsets of A which are congruent by translation. Of course, this also implies that n disjoint k-element subsets of A exist which are congruent by translation. This improves an earlier result of the first author for k = n = 2 on B_{2}-sequences, which was published in a paper of *A.Stöhr* [J. Reine Angew. Math. 194, 111-140 (1955; Zbl 066.03101)].

One obtains the corollary that for every two natural numbers k,n the set of prime numbers has k disjoint n-element subsets which are congruent by translation. Concerning the sharpness of the theorem there holds: If 0 < \alpha < 1-1/k-1/n+1/kn then for all sufficiently large natural numbers m there exists a subset of **{**1,...,m**}** which has at least m^{\alpha} elements but no k disjoint congruent n-element subsets.

**Classif.: ** * 11B83 Special sequences of integers and polynomials

11B05 Topology etc. of sets of numbers

**Keywords: ** disjoint n-element subsets; congruent by translation

**Citations: ** Zbl 066.031

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