On sums of subsets of a set of integers

For r≧2 let p(n, r) denote the maximum cardinality of a subset A of N={1, 2,..., n} such that there are no B⊂A and an integer y with {Mathematical expression}b=y^r. It is shown that for any ε>0 and n>n(ε), (1+o(1))2^1/(r+1)n^(r-1)/(r+1)≦p(n, r)≦n^e{open}+2/3 for all r≦5, and that for every fixed r≧6, p(n, r)=(1+o(1))·2^1/(r+1)n^(r-1)/(r+1) as n→∞. Let f(n, m) denote the maximum cardinality of a subset A of N such that there is no B⊂A the sum of whose elements is m. It is proved that for 3 n^6/3+e{open}≦m≦n²/20 log²n and n>n(ε), f(n, m)=[n/s]+s-2, where s is the smallest integer that does not divide m. A special case of this result establishes a conjecture of Erdo{combining double acute accent}s and Graham.