20.500.12556/RUP-22140 Optimal bounds for zero-sum cycles. I. Odd order Optimal bounds for zero-sum cycles 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. Zero-sum Ramsey theory directed cycles Zero-sum cycles Ramseyjeva teorija ničelne vsote usmerjeni cikli cikli ničelne vsote true false true Angleški jezik Slovenski jezik Članek v reviji 2025-11-24 14:57:24 2025-11-24 14:57:26 2025-11-25 03:06:03 0000-00-00 00:00:00 2025 0 0 str. 246-256 Vol. 173 Jul. 2025 0000-00-00 Zaloznikova NiDoloceno NiDoloceno 0000-00-00 0000-00-00 2025-04-16 51 0095-8956 10.1016/j.jctb.2025.04.003 258490115 RAZ_Campbell_Rutger_2025.pdf RAZ_Campbell_Rutger_2025.pdf 1 04D3C761C9D4A49FC0F15D068A107C11 0c33be95bdeabf9d64e380f8162e7d386c11da8cfa5072401aba6e4ebcfe99b2 e528af88-c93d-11f0-8f0b-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=32125 https://www.sciencedirect.com/science/article/pii/S0095895625000243?via%3Dihub 1 c39f2370-c93d-11f0-8f0b-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=32124 Fakulteta za matematiko, naravoslovje in informacijske tehnologije 0 0 0