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 \(\sum\limits_{i\in b} a_i = \sum\limits_{i\in b} b_i = 0\).