  Let $G$ be a finite abelian group, and let $m>0$ with $\exp(G)\mid m$. Let
$s_{m}(G)$ be the generalized Erd\H{o}s-Ginzburg-Ziv invariant which denotes
the smallest positive integer $d$ such that any sequence of elements in $G$ of
length $d$ contains a subsequence of length $m$ with sum zero in $G$. For any
integer $r>0$, let $\mathcal{I}_m^{(r)}$ be the collection of all $r$-uniform
intersecting families of size $m$. Let $R(\mathcal{I}_m^{(r)},G)$ be the
smallest positive integer $d$ such that any $G$-coloring of the edges of the
complete $r$-uniform hypergraph $K_{d}^{(r)}$ yields a zero-sum copy of some
intersecting family in $\mathcal{I}_m^{(r)}$. Among other results, we mainly
prove that $\Omega(s_{m}(G))-1\leq R (\mathcal{I}_{m}^{(r)}, \ G)\leq
\Omega(s_{m}(G)),$ where $\Omega(s_{m}(G))$ denotes the least positive integer
$n$ such that ${n-1 \choose r-1}\geq s_{m}(G)$, and we show that if $r\mid
\Omega(s_{m}(G))-1$ then $R (\mathcal{I}_{m}^{(r)}, \ G)= \Omega(s_{m}(G))$.
