On sums of subsets of a set of integers


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.

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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.

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

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