AI 中文总结
本文围绕树 - 圈拉姆齐数展开，Burr等人曾询问满足特定条件的最小函数\(f(m)\)，前人不断改进其上限，本文用不同方法证明\(f(m)\leq2m - 4\)，并提出\(f(m)=\lceil (2m - 1)/3\rceil\)的猜想。
AI 中文摘要
设\(R(T_n,C_m)\)表示\(n\)个顶点的树\(T_n\)与长度为\(m\)的圈\(C_m\)的拉姆齐数。Burr等人（1982年）询问最小函数\(f(m)\)，使得当\(n\geq f(m)\)且\(m\geq3\)为奇数时，\(R(T_n,C_m)=2n - 1\)。他们证明\(f(m)\leq756m^{10}\)，后经Brennan（2016年）改进为\(25m\)，Fan和Lin（2025年）改进为\(4m - 8\)。本文用不同方法证明\(f(m)\leq2m - 4\)，并猜想\(f(m)=\lceil (2m - 1)/3\rceil\)。
英文摘要
Let $R(T_n,C_m)$ denote the Ramsey number of a tree $T_n$ on $n$ vertices versus a cycle $C_m$ of length $m$. Burr, Erdős, Faudree, Rousseau, and Schelp (1982) asked for the least function $f(m)$ such that $R(T_n,C_m)=2n-1$ for every odd $m\ge 3$ whenever $n\ge f(m)$. They proved that $f(m)\le 756m^{10}$. This bound was later improved to $25m$ by Brennan (2016) and to $4m-8$ by Fan and Lin (2025). In this note, we show that $f(m)\le 2m-4$ by using a different method and conjecture that $f(m)=\lceil (2m-1)/3\rceil$.
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