奇圈的拉姆齐数的新上界
New upper bound for the Ramsey number of odd cycles
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中文总结 AI 辅助
本文针对固定$\ell\geq2$且$k$足够大的情况，给出了奇圈拉姆齐数$R_k(C_{2\ell+1})$的新上界，大幅改进了Axenovich等人和Miyazaki等人的已有结果。
中文摘要 AI 辅助
$k$色拉姆齐数$R_k(C_{2\times\ell+1})$是指最小的整数$n$，使得完全图$K_n$的任意$k$边着色都包含单色奇圈$C_{2\ell+1}$。Axenovich、Cames van Batenburg、Janzer、Michel和Rundström（JCT-B，2026）近期证明$R_k(C_{2\ell+1})\leq (4\ell-2)^k k^{k/\ell}+1$，Miyazaki等人进一步将因子$k^{k/\ell}$改进为$(k!)^{1/\ell}$。由于Jenssen和Skokan（AM，2021）确定了固定$k$且$\ell$足够大时的$R_k(C_{2\ell+1})$，因此寻求固定$\ell$且$k$足够大时的更优界变得更有意义。本文证明，对每个固定的$\ell\geq2$且$k$足够大，$R_k(C_{2\ell+1})\leq \frac{2\ell}{2\ell-1}(2\ell-1)^k(k!)^{1/\ell}\exp\left(k^{1-1/\ell}+O_\ell\left(k^{1-2/\ell}+\log k\right)\right)+1$，该结果将Miyazaki等人的界改进了$2^{k-o(k)}$倍，将Axenovich等人的界改进了$(2e^{1/\ell})^{k-o(k)}$倍。
英文摘要
The \emph{$k$-color Ramsey number} $R_k(C_{2\ell+1})$ is the least integer $n$ such that any $k$-edge-coloring of a complete graph $K_n$ has a monochromatic odd cycle $C_{2\ell+1}$. Axenovich, Cames van Batenburg, Janzer, Michel, and Rundström~(JCT-B, 2026) recently proved \[ R_k(C_{2\ell+1})\le (4\ell-2)^k k^{k/\ell}+1, \] and Miyazaki, Mulrenin, Pohoata, and Zheng further improved the factor $k^{k/\ell}$ to $(k!)^{1/\ell}$. As Jenssen and Skokan (AM, 2021) determined $R_k(C_{2\ell+1})$ for fixed $k$ and sufficiently large $\ell$, it becomes even more interesting to seek better bound for fixed $\ell$ and sufficiently large $k$. In this paper, we show \[ R_k(C_{2\ell+1}) \le \frac{2\ell}{2\ell-1}(2\ell-1)^k(k!)^{1/\ell} \exp\!\left(k^{1-1/\ell}+O_\ell\!\left(k^{1-2/\ell}+\log k\right)\right)+1 \] for every fixed $\ell\ge 2$ 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 $(2\e^{1/\ell})^{k-o(k)}$.
