Topological methods in zero-sum Ramsey theory

A landmark result of Erdős, Ginzburg, and Ziv (EGZ) states that any sequence of $2n-1$ elements in 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 is a topological criterion for determining when any 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 give a fractional generalization of the EGZ theorem with applications to balanced set families and provide a constrained EGZ theorem which imposes combinatorial restrictions on zero-sum sequences in the original result.