##
**Zentralblatt MATH**

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

**Zbl.No: ** 725.11012

**Autor: ** Erdös, Paul; Sárközy, A.

**Title: ** On a problem of Straus. (In English)

**Source: ** Disorder in physical systems, Vol. in Honour of J. M. Hammersley 70th Birthday, 55-66 (1990).

**Review: ** [For the entire collection see Zbl 714.00019.]

If *A* is a set of integers with the property that no element a_{i} is the average of any subset of *A* consisting of two or more elements, then *A* is said to be non-averaging. Let f(N) denote the maximum cardinality of a non-averaging subset of **{**0,1,2,...,N**}**. The best estimates for f(N) are due to *Á.P.Bosznay* (1989; Zbl 682.10049) and *P.Erdös* and *E.G.Straus* (1970; Zbl 216.01503) who showed that f(N) >> N^{1/4} and f(N) << N^{2/3} respectively. In this paper, the authors improve this last estimate to: For N > N_{0}, f(N) < 403(N log N)^{ ½}.

Denote by *P*(*A*) the set of distinct integers n which can be represented in the form n = **sum**_{a in A}\epsilon_{a}a where \epsilon_{a} = 0 or 1 for all a and 0 < **sum**_{a in A}\epsilon_{a} < oo. The above estimate for f(N) is obtained via a bound for F(N) which is defined to be the largest k such that there exist two subsets *A* = **{**a_{1},...,a_{k}**}**, *B* = **{**b_{1},...,b_{k}**}** of **{**0,1,...,N**}** with *P*(*A*)\cap *P*(*B*) = Ø. The infinite analogue of this problem is: If *A*,*B* are infinite sets of positive integers with *P*(*A*)\cap *P*(*B*) = Ø then how large can **max** (A(x),B(x)) be? Here A(x), B(x) are the counting functions of *A*,*B* respectively. The authors conjecture that **liminf**_{x ––> oo}\frac{**max** (A(x),B(x))}{x^{ ½}} = 0 and show by an interesting construction that the x^{ ½} in the conjecture cannot be replaced by x^{ ½}(log x)^{- ½-\epsilon}.

**Reviewer: ** M.Nair (Glasgow)

**Classif.: ** * 11B99 Sequences and sets

05D05 Extremal set theory

00A07 Problem books

**Keywords: ** non-averaging sets; maximum cardinality

**Citations: ** Zbl 714.00019

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag