On the Erd\H{o}s-Ginzburg-Ziv invariant and zero-sum Ramsey number for intersecting families

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}${m}^{®}, \ G)\leq \Omega(s{m}(G)),$where$\Omega(s_{m}(G))$denotestheleastpositiveinteger$n$suchthat${n-1 \choose r-1}\geq s_{m}(G)$,andweshowthatif$r\mid \Omega(s_{m}(G))-1$then$R ($\mathcal{I}${m}^{®}, \ G)= \Omega(s{m}(G))$.