Zero-sum subsets in vector spaces over finite fields

The Olson constant OL(F-p(d)) represents the minimum positive integer t with the property that every subset A subset of F-p(d) of cardinality t contains a nonempty subset with vanishing sum. The problem of estimating OL(F-p(d)) is one of the oldest questions in additive combinatorics, with a long and interesting history even for the case d = 1. We prove that for any fixed d >= 2 and epsilon > 0, the Olson constant of F-p(d) satisfies the inequality OL(F-p(d)) <= (d - 1+ epsilon) p for all sufficiently large primes p. This settles a conjecture of Hoi Nguyen and Van Vu.