Subset sums

Suppose ε > 0 and k > 1. We show that if n > n₀(k, ε) and A ⊆ Z_n satisfies |A| > (( 1 k) + ε)n then there is a subset B ⊆ A such that 0 < |B| ≤ k and Σ_b∈Bb = 0 (in Z_n). The case k = 3 solves a problem of Stalley and another problem of Erdös and Graham. For an integer m > 0, let snd(m) denote the smallest integer that does not divide m. We prove that for every ε > 0 there is a constant c = c(ε) > 1, such that for every n > 0 and every m, n^{1 + ε} ≤ m ≤ n² log²n every set A ⊆ {1, 2,..., n} of cardinality |A| > c · n snd(m) contains a subset B ⊆ A so that Σ_b∈Bb = m. This is best possible, up to the constant c. In particular it implies that for every n there is an m such that every set A ⊆ {1,...,n} of cardinality |A| > cn log n contains a subset B ⊆ A so that Σ_b∈Bb = m, thus settling a problem of Erdös and Graham.