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 a1 < ... < ak \leq n such that all ai+aj 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+n1/4+1 [J. Lond. Math. Soc. 16, 212-215 (1941; Zbl 061.07301)]. Now let m,n1,...,nm, c be positive integers, let A = {A1,...,Am} be a system of sequences of integers Ai = {ai,1 < ... < ai,ni}, i = 1,...,m, and let Di = {ai,j-ai,k| 1 \leq k < j \leq ni} be the difference set of Ai. The system S = {D1,...,Dm} is called perfect for c if the set D = \cupi = 1mDi consists of the integers t such that c \leq t \leq c-1+N where N = sumi = 1n\binom{ni}{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 ai,j-ai,k in each Di are all distinct and in [1,N] and if Di1\cap Di2 = Ø for all 1 \leq i1 < i2 \leq m, then for every \epsilon > 0 there is an \eta > 0 so that, for N > N0(\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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