Let $[n]$ denote the subset of $2n+1$ integers with absolut value at most $n$.
Consider $k$ (not necessarily different) vectors in $[n]^d$ summing up to zero. What is the smallest $k>1$ (for fixed $n$ and $d$) that assures the existence of a strict subsequence of vectors also summing up to zero?


I am most interested in the case $n=1$.



Zero subsums of integer vectors
For n=1 we have recently solved the problem, the answer is around $d^d$, see <http://arxiv.org/abs/0912.0424> By we I mean my coauthors and by solve I mean gave a reasonable lower and upper bound.


