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

TL;DR

This paper explores the relationship between the Erd ext{"o}s-Ginzburg-Ziv invariant and zero-sum Ramsey numbers for intersecting families in hypergraphs, establishing bounds and exact values under certain divisibility conditions.

Contribution

It provides bounds and exact values for zero-sum Ramsey numbers related to intersecting families in hypergraphs, linking them to the Erd ext{"o}s-Ginzburg-Ziv invariant.

Findings

Bounds for R(_{m}^{(r)}, G) in terms of s_{m}(G)

Exact value of R(_{m}^{(r)}, G) when r divides _{m}^{(r)}-1

Connection between hypergraph coloring and zero-sum subsequences

Abstract

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 $R (\mathcal{I}_{m}^{(r)}, \ G)= \Omega(s_{m}(G))$.