| Title: | Optimal bounds for zero-sum cycles. I. Odd order |
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| Authors: | ID Campbell, Rutger (Author)
ID Gollin, J. Pascal (Author)
ID Hendrey, Kevin (Author)
ID Steiner, Raphael (Author) |
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| Files: |  RAZ_Campbell_Rutger_2025.pdf (398,18 KB)
MD5: 04D3C761C9D4A49FC0F15D068A107C11
 https://www.sciencedirect.com/science/article/pii/S0095895625000243?via%3Dihub
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| Language: | English |
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| Work type: | Article |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | FAMNIT - Faculty of Mathematics, Science and Information Technologies
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| Abstract: | For a finite (not necessarily abelian) group Γ, let n(Γ) denote the smallest positive integer $n$ such that for each labelling of the arcs of the complete digraph of order n using elements from Γ, there exists a directed cycle such that the arc-labels along the cycle multiply to the identity. Alon and Krivelevich [2] initiated the study of the parameter n(.) on cyclic groups and proved n(Z_q) = O(q log q). This was later improved to a linear bound of n(Γ) <= 8|Γ| for every finite abelian group Γ by Mészáros and the last author [8], and then further to n(Γ) <= 2|Γ|-1 for every non-trivial finite group independently by Berendsohn, Boyadzhiyska and Kozma [3] as well as by Akrami, Alon, Chaudhury, Garg, Mehlhorn and Mehta [1]. In this series of two papers we conclude this line of research by proving that n(Γ) < |Γ|+1 for every finite group Γ, which is the best possible such bound in terms of the group order and precisely determines the value for all cyclic groups as n(Z_q) = q+1. In the present paper we prove the above result for all groups of odd order. The proof for groups of even order needs to overcome substantial additional obstacles and will be presented in the second part of this series. |
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| Keywords: | Zero-sum Ramsey theory, directed cycles, Zero-sum cycles |
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| Publication version: | Version of Record |
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| Publication date: | 16.04.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | str. 246-256 |
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| Numbering: | Vol. 173 |
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| PID: | 20.500.12556/RUP-22140  |
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| UDC: | 51 |
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| ISSN on article: | 0095-8956 |
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| DOI: | 10.1016/j.jctb.2025.04.003 |
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| COBISS.SI-ID: | 258490115 |
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| Publication date in RUP: | 24.11.2025 |
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| Views: | 293 |
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| Downloads: | 4 |
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