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arXiv · 2310.17065

Topological methods in zero-sum Ramsey theory

Florian Frick·Jacob Lehmann Duke·Meenakshi McNamara·Hannah Park-Kaufmann·Steven Raanes·Steven Simon·Darrion Thornburgh·Zoe Wellner
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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.

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Florian Frick, Jacob Lehmann Duke, Meenakshi McNamara, Hannah Park-Kaufmann, Steven Raanes, Steven Simon, Darrion Thornburgh, Zoe Wellner. 2023-10-25. Topological methods in zero-sum Ramsey theory. https://doi.org/10.1017/fms.2025.10125

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