On sums of subsets of a set of integers

Abstract

Forr≧2 letp(n, r) denote the maximum cardinality of a subsetA ofN={1, 2,...,n} such that there are noB⊂A and an integery with\(\mathop \sum \limits_{b \in B} b = y^r \) b=y r. It is shown that for anyε>0 andn>n(ε), (1+o(1))21/(r+1) n (r−1)/(r+1)p(n, r)≦n ɛ+2/3 for allr≦5, and that for every fixedr≧6,p(n, r)=(1+o(1))·21/(r+1) n (r−1)/(r+1) asn→∞. Letf(n, m) denote the maximum cardinality of a subsetA ofN such that there is noB⊂A the sum of whose elements ism. It is proved that for 3n 6/3+ɛmn 2/20 log2 n andn>n(ε), f(n, m)=[n/s]+s−2, wheres is the smallest integer that does not dividem. A special case of this result establishes a conjecture of Erdős and Graham.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    N. Alon, Subset sums,J. Number Theory,27 (1987), 196–205.

    Article  Google Scholar 

  2. [2]

    P.Erdős and G.Freiman, On two additive problems,J. Number Theory, to appear.

  3. [3]

    P.Erdős, Some problems and results on combinatorial number theory,Proc. First China Conference in Combinatorics (1986),to appear.

  4. [4]

    E.Lipkin, On representation ofr-powers by subset sums,Acta Arithmetica, submitted.

  5. [5]

    J. Olson, An additive theorem modulop, J. Combinatorial Theory 5 (1968), 45–52.

    Google Scholar 

  6. [6]

    E. C. Titchmarsh,Introduction to the theory of Fourier integrals, Oxford at the Clarendon Press, London, 1948, p. 177.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

Research supported in part by Allon Fellowship, by a Bat-Sheva de Rothschild Grant and by the Fund for Basic Research administered by the Israel Academy of Sciences.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Alon, N., Freiman, G. On sums of subsets of a set of integers. Combinatorica 8, 297–306 (1988). https://doi.org/10.1007/BF02189086

Download citation

AMS subject classification (1980)

  • 10 A 50
  • 10 B 35
  • 10 J 10