阈值Ramsey重数及奇圈的极值着色

中文摘要 AI 辅助

图$H$的Ramsey数$r(H)$是最小正整数$n$，使得在$n$个顶点的完全图$K_n$的任意红/蓝边着色中，都存在一个单色的$H$副本。图$H$的阈值Ramsey重数$m(H)$是在$K_{r(H)}$的所有红/蓝边着色中，单色$H$副本的最小数量。唯一已确定$m(H)$的图族是星图，由Harary和Prins（1974）给出。设$C_k$为$k$个顶点的圈。Conlon、Fox、Sudakov和Wei（2022）猜想：对每个足够大的奇数$k$，$m(C_k)=(k-1)!/2$。本文证实了这一猜想，并刻画了$K_{2k-1}$的所有极值着色。这也是第二个已确定$m(H)$的图族。

英文摘要

The Ramsey number $r(H)$ of a graph $H$ is the minimum positive integer $N$ such that every red/blue edge-coloring of the complete graph $K_N$ on $N$ vertices contains a monochromatic copy of $H$. The Ramsey multiplicity $M(H,n)$ is the minimum number of monochromatic copies of $H$ over all red/blue edge-colorings of $K_n$. It is called threshold Ramsey multiplicity if $n=r(H)$, and denoted by $m(H)$. The only previously known general infinite family for which $m(H)$ has been determined is stars, due to Harary and Prins (1974). Let $C_k$ denote a cycle on $k$ vertices. Conlon, Fox, Sudakov, and Wei (2022) conjectured that $m(C_k)=(k-1)!/2$ for every sufficiently large odd integer $k$. In this paper, we determine $M(C_k,r(C_k)+\ell)$ for every fixed nonnegative integer $\ell$ and all sufficiently large odd $k$, and characterize all extremal colorings, thereby confirming the conjecture. This is also a second general infinite family for which $m(H)$ has been determined.

发表机构

• School of Mathematics, Nanjing University（南京大学数学学院）
• School of Mathematics and Statistics, Nanjing University of Science and Technology（南京理工大学数学与统计学院）

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