On the Erdős-Ginzburg-Ziv invariant and zero-sum Ramsey number for intersecting families

Abstract

Let $G$ be a finite abelian group, and let $m>0$ with $\exp (G)\mid m$. Let $s_m(G)$ be the generalized Erdős-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 $\Omega (s_m(G))-1\le R({\mathcal{I}}_m^{(r)},G)\le \Omega (s_m(G)),$ where $\Omega (s_m(G))$ denotes the least positive integer $n$ such that $\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)\ge s_m(G)$, and we show that if $r\mid \Omega (s_m(G))-1$ then $R({\mathcal{I}}_m^{(r)},G)=\Omega (s_m(G))$.

arXiv comment14 papes