On the Intersection of Two $m$-sets and the Erdos-Ginzburg-Ziv Theorem

Abstract

We prove the following extension of the Erdős-Ginzburg-Ziv Theorem. Let $m$ be a positive integer. For every sequence $\{a_i{\}}_{i\in I}$ of elements from the cyclic group ${\mathbb{Z}}_m$, where $|I|=4m–5$ (where $|I|=4m–3$), there exist two subsets $A,B\subseteq I$ such that $|A\cap B|=2$ (such that $|A\cap B|=1$), $|A|=|B|=m$, and $\sum _{i\in b}{a}_i=\sum _{i\in b}{b}_i=0$.