  The Olson constant $\mathcal{O}L(\mathbb{F}_{p}^{d})$ represents the minimum
positive integer $t$ with the property that every subset $A\subset
\mathbb{F}_{p}^{d}$ of cardinality $t$ contains a nonempty subset with
vanishing sum. The problem of estimating $\mathcal{O}L(\mathbb{F}_{p}^{d})$ is
one of the oldest questions in additive combinatorics, with a long and
interesting history even for the case $d=1$.
  In this paper, we prove that for any fixed $d \geq 2$ and $\epsilon > 0$, the
Olson constant of $\mathbb{F}_{p}^{d}$ satisfies the inequality
$$\mathcal{O}L(\mathbb{F}_{p}^{d}) \leq (d-1+\epsilon)p$$ for all sufficiently
large primes $p$. This settles a conjecture of Hoi Nguyen and Van Vu.

Combinatorics;Number Theory