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 $n=1$.