2023/10/25 by Florian Frick, Jacob Lehmann Duke, Frick, Florian +13
Computer Science · Mathematics · #05C55 #05E16 #55M20 #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2310.17065
openalex publication_date 2023/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A cornerstone result of Erd\H os, 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 (1) 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 (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.