The Erdős–Ginzburg–Ziv Theorem and Zero-Sum Problems in Additive Combinatorics

Abstract

This study explores the Erdős–Ginzburg–Ziv (EGZ) theorem, a cornerstone result in additive combinatorics that reveals the inherent arithmetic structure imposed by combinatorial constraints. The theorem asserts that any sequence of 2n−1 integers contains an n-term subsequence whose sum is divisible by n, establishing one of the most elegant examples of a forced zero-sum configuration. Building on this foundation, the study examines the broader landscape of zero-sum problems, focusing on key invariants such as the Davenport constant and the EGZ constant, which measure the minimal conditions guaranteeing zero-sum subsequences in various finite Abelian groups. Modern developments—including the polynomial method, weighted zero-sum extensions, and investigations into non-Abelian settings—are highlighted for their role in expanding the applicability of EGZ-type results. Applications in coding theory, additive number theory, and algebraic geometry demonstrate the theorem’s interdisciplinary influence. Finally, the study outlines major open problems, including determining the EGZ constant for general groups and classifying extremal or near-extremal sequences, underscoring the continuing significance of zero-sum phenomena. Altogether, the EGZ theorem serves not only as a historical milestone but as a catalyst for ongoing research in combinatorics, algebra, and beyond.