Topological methods in zero-sum Ramsey theory

TL;DR

This paper introduces topological methods into zero-sum Ramsey theory, providing new criteria and generalizations for zero-sum problems in hypergraphs and finite groups, extending classical results like the EGZ theorem.

Contribution

It develops a topological criterion for zero-sum hyperedges, recovers Olson's generalization, and introduces fractional and constrained versions of the EGZ theorem.

Findings

Topological criterion for zero-sum hyperedges in hypergraphs

Recovery of Olson's generalization of EGZ theorem

Fractional and constrained generalizations of EGZ theorem

Abstract

A cornerstone result of Erd\H os, Ginzburg, and Ziv (EGZ) states that any sequence of $2n-1$ elements in $\mathbb{Z}/n$ contains a zero-sum subsequence of length $n$. While algebraic techniques have predominated in deriving many deep generalizations of this theorem over the past sixty years, here we introduce topological approaches to zero-sum problems which have proven fruitful in other combinatorial contexts. Our main result (1) is a topological criterion for determining when any $\mathbb{Z}/n$-coloring of an $n$-uniform hypergraph contains a zero-sum hyperedge. In addition to applications for Kneser hypergraphs, for complete hypergraphs our methods recover Olson's generalization of the EGZ theorem for arbitrary finite groups. Furthermore, we (2) give a fractional generalization of the EGZ theorem with applications to balanced set families and (3) provide a constrained EGZ theorem which imposes combinatorial restrictions on zero-sum sequences in the original result.