Title: Threshold Ramsey multiplicity for odd cycles Show Chinese title

Title: 奇圈的阈值 Ramsey 重数

Abstract: The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic copies of $H$ taken over all two-edge-colorings of $K_{r(H)}$. The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant $c$ such that the threshold Ramsey multiplicity for a path or even cycle with $k$ vertices is at least $(ck)^k$, which is tight up to the value of $c$. Here, using different methods, we show that the same result also holds for odd cycles with $k$ vertices. Show Chinese abstract

Abstract: 图 $H$ 的 Ramsey 数 $r(H)$ 是最小的 $n$，使得任何完全图 $K_n$ 的边的二色着色都包含一个 $H$ 的单色副本。 阈值Ramsey多重性$m(H)$是所有二边着色下的单色$H$的最小数目，这些二边着色是针对$K_{r(H)}$的。 这一概念的研究最早由Harary和Prins在近五十年前提出。 在一篇配套论文中，作者证明存在一个正的常数$c$，使得具有$k$个顶点的路径或偶数循环的阈值Ramsey多重性至少为$(ck)^k$，这在$c$的值上是紧致的。 这里，使用不同的方法，我们证明了对于具有$k$个顶点的奇数环，同样的结果也成立。