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**Zentralblatt MATH**

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

**Zbl.No: ** 491.10044

**Autor: ** Erdös, Paul

**Title: ** Some problems on additive number theory. (In English)

**Source: ** Ann. Discrete Math. 12, 113-116 (1982).

**Review: ** Let f(n) be the largest integer k for which there is a sequence 1 \leq a_{1} < ... < a_{k} \leq n such that all a_{i}+a_{j} are distinct. The author and *P.Turán* have conjectured that f(n) = n^{ ½}+O(1) and have proved that n^{ ½}-n^{ ½-c} < f(n) < n^{ ½}+n^{1/4}+1 [J. Lond. Math. Soc. 16, 212-215 (1941; Zbl 061.07301)]. Now let m,n_{1},...,n_{m}, c be positive integers, let A = **{**A_{1},...,A_{m}**}** be a system of sequences of integers A_{i} = **{**a_{i,1} < ... < a_{i,ni}**}**, i = 1,...,m, and let D_{i} = **{**a_{i,j}-a_{i,k}| 1 \leq k < j \leq n_{i}**}** be the difference set of A_{i}. The system S = **{**D_{1},...,D_{m}**}** is called perfect for c if the set D = \cup_{i = 1}^{m}D_{i} consists of the integers t such that c \leq t \leq c-1+N where N = **sum**_{i = 1}^{n}\binom{n_{i}}{2}. J.Abrham has proved [Ann. Discrete Math. 12, 1-7 (1982)] that, for every perfect system, m > \alpha N where \alpha > 0 is an absolute constant. The method that the author and Turán used to obtain their upper bound for f(n) is now used to t if the integers a_{i,j}-a_{i,k} in each D_{i} are all distinct and in [1,N] and if D_{i1}\cap D_{i2} = Ø for all 1 \leq i_{1} < i_{2} \leq m, then for every \epsilon > 0 there is an \eta > 0 so that, for N > N_{0}(\epsilon,\eta), if |D| > (1+\epsilon)N/2 then m > \eta N. This theorem is then used to establish Abrham's result.

**Reviewer: ** B.Garrison

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

05B10 Difference sets

11B75 Combinatorial number theory

**Keywords: ** distinct sums; perfect systems of difference sets

**Citations: ** Zbl.061.073

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