New upper bound for the Ramsey number of odd cycles

2026/08/03 by Ting Huang, Jiabao Yang, Yaojun Chen
Mathematics · #math.CO #msc:05C55 #msc:05C38 #msc:05C15

paper · pdf

13 pages

arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

The k-color Ramsey number R_k(C_2ℓ+1) is the least integer n such that any k-edge-coloring of a complete graph K_n has a monochromatic odd cycle C_2ℓ+1. Axenovich, Cames van Batenburg, Janzer, Michel, and Rundström~(JCT-B, 2026) recently proved R_k(C_2ℓ+1)≤ (4ℓ-2)^k k^k/ℓ+1, and Miyazaki, Mulrenin, Pohoata, and Zheng further improved the factor k^k/ℓ to (k!)^1/ℓ. As Jenssen and Skokan (AM, 2021) determined R_k(C_2ℓ+1) for fixed k and sufficiently large ℓ, it becomes even more interesting to seek better bound for fixed ℓ and sufficiently large k. In this paper, we show R_k(C_2ℓ+1) ≤ (2ℓ)/(2ℓ-1)(2ℓ-1)^k(k!)^1/ℓ exp (k^1-1/ℓ+O_ℓ (k^1-2/ℓ+log k))+1 for every fixed ℓ≥ 2 and sufficiently large k, which improves the bound of Miyazaki et al. by a factor 2^k-o(k), and the bound of Axenovich et al. by a factor (2\e^1/ℓ)^k-o(k).

Citations

• An improved upper bound for the multicolour Ramsey number of odd cycles
• New lower bounds for Schur and weak Schur numbers
• Exact Ramsey numbers of odd cycles via nonlinear optimisation
• Multicolour Ramsey Numbers of Odd Cycles
• The Cycle-Complete graph Ramsey numbers