Topological methods in zero-sum Ramsey theory
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arXiv
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| Autori principali: | , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| author | Frick, Florian Duke, Jacob Lehmann McNamara, Meenakshi Park-Kaufmann, Hannah Raanes, Steven Simon, Steven Thornburgh, Darrion Wellner, Zoe |
| author_facet | Frick, Florian Duke, Jacob Lehmann McNamara, Meenakshi Park-Kaufmann, Hannah Raanes, Steven Simon, Steven Thornburgh, Darrion Wellner, Zoe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_17065 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Topological methods in zero-sum Ramsey theory Frick, Florian Duke, Jacob Lehmann McNamara, Meenakshi Park-Kaufmann, Hannah Raanes, Steven Simon, Steven Thornburgh, Darrion Wellner, Zoe Combinatorics Algebraic Topology 05C55, 05E16, 55M20 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. |
| title | Topological methods in zero-sum Ramsey theory |
| topic | Combinatorics Algebraic Topology 05C55, 05E16, 55M20 |
| url | https://arxiv.org/abs/2310.17065 |