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))2^1/(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))·2^1/(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+ɛ≦m≦n 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.