发表机构
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了Pokrovskiy等人关于圈与图的拉姆齐数的猜想，证明了满足线性条件时拉姆齐数取Burr给出的最优下界，给出了首个线性依赖于图规模的最优界。
AI 中文摘要
对于图F和H，拉姆齐数R(F,H)是满足每个N顶点图都包含F或其补图包含H的最小整数N。若F连通且|F|≥σ(H)，Burr的构造给出R(F,H)≥(χ(H)-1)(|F|-1)+σ(H)，其中σ(H)表示H的正常χ(H)着色中最小色类的阶数。Burr证明当n足够大时，该界对F=Cn成立；Allen、Brightwell和Skokan猜想当n≥|H|χ(H)时等式已成立，Haslegrave、Hyde、Kim和Liu随后证明当n≥C|H|log⁴χ(H)时成立。Pokrovskiy和Sudakov猜想最优线性条件n≥C|H|即可，该猜想也被Montgomery在其2026年国际数学家大会（ICM）综述中强调（见猜想9.2）。本文通过证明存在绝对常数C>0，使得对每个非空图H和每个n≥C|H|，均有R(Cn,H)=(χ(H)-1)(n-1)+σ(H)，从而解决了该猜想。这是首个线性依赖于|H|的界，且在常数因子范围内是最优的。我们的证明基于Haslegrave、Hyde、Kim和Liu的框架，结合了扩张性与切换圈长度的新思想。
英文摘要
For graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the minimum integer $N$ such that every $N$-vertex graph contains $F$ or its complement contains $H$. If $F$ is connected and $|F|\geσ(H)$, a construction of Burr gives $R(F,H)\ge(χ(H)-1)(|F|-1)+σ(H)$, where $σ(H)$ denotes the minimum order of a color class in a proper $χ(H)$-coloring of $H$. Burr proved that this bound is attained for $F=C_n$ when $n$ is sufficiently large. Allen, Brightwell and Skokan conjectured that equality already holds whenever $n\geq |H| χ(H)$, while Haslegrave, Hyde, Kim and Liu subsequently proved it whenever $n\ge C|H|\log^4χ(H)$. Pokrovskiy and Sudakov conjectured that the optimal linear condition $n\geq C|H|$ suffices; this conjecture was also highlighted by Montgomery in his 2026 ICM survey (see Conjecture 9.2). In this paper, we resolve this conjecture by proving that there is an absolute constant $C>0$ such that $R(C_n,H)=(χ(H)-1)(n-1)+σ(H)$ for every nonempty graph $H$ and every $n\ge C|H|$. This gives the first bound linear in $|H|$, and is best possible up to a constant factor. Our proof builds on the framework of Haslegrave, Hyde, Kim and Liu, and combines some new ideas in expansion and switching cycle lengths.
