Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  216.01503
Autor:  Erdös, Paul; Straus, E.G.
Title:  Nonaveraging sets. II (In English)
Source:  Combinat. Theory Appl., Colloquia Math. Soc. Janos Bolyai 4, 405-411 (1970).
Review:  [For the entire collection see Zbl 205.00201.]
We wish to consider sets of integers A = {a1, ... ,an} so that 0 \leq a1 < a2 < ... < an \leq x and no ai is the arithmetic mean of any subset of A consisting of two or more elements. In Part I [by the second author in Proc. Sympos. pure Math., Am. Math. Soc. (1967)] is initiated the study of the maximal number of elements in nonaveraging sets and sets which satisfy related conditions. Using the notation of Part I we define f(x) as the maximal number of elements in a nonaveraging set; h(x) as the number of elements of a maximal set of integers in the interval [0,x] such that no two distinct subsets have the same arithmetic mean; and h^*(x) as the number of elements of a maximal set of integers in [0,x] such that no two subsets with a relatively prime number of elements have the same arithmetic mean. In Part I we proved (logrx = log x/ log r):

log2 f(x) > \sqrt{2 log2x}+ 1/2 +O(1/\sqrt{log 2}),

(1+\sigma(1)) log x/ log log x < h(x) < log2x+O(log log x),     (*)

log2h^*(x) \geq \sqrt{log2x}-1+0(1/ \sqrt{log x}).

In the present note we prove in \S2 that (*) can be replaced by

-1+ log4x \leq h(x) < log2x+O(log log x).

Next, in \S3, we prove that even if we ease the restriction on our sets so that only subsets with different numbers of elements must have different averages then the maximal number, h**(x), of elements satisfies

h**(x) < c(log x)2 for some constant c.

Finally in \S4 we get an upper bound for f(x) < cx3/4.
Classif.:  * 05A99 Classical combinatorial problems
Citations:  Zbl 205.00201(EA)

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