Remarks on some zero-sum problems

Let Z denote the ring of integers and for a prime p and positive integers r and d, let fr(P, d) denote the smallest positive integer such that given any sequence of fr(p, d) elements in (Z/pZ(d, there exists a subsequence of (rp) elements whose sum is zero in (Z/pZ(d. That f1(p, 1) = 2p − 1, is a classical result due to Erdős, Ginzburg and Ziv. Whereas the determination of the exact value of f1(p, 2) has resisted the attacks of many well known mathematicians, we shall see that exact values of fr(p, 1) for r ≥ 1 can be easily obtained from the above mentioned theorem of Erdős, Ginzburg and Ziv and those of fr(p, 2) for r ≥ 2 can be established by the existing techniques developed by Alon, Dubiner and Rónyai in connection with obtaining good upper bounds for f1(p, 2). We shall also take this opportunity to describe some of the early results in the introduction.