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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.
The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. While this definition only asks for a single monochromatic copy of $H$, it is
For graphs $G$ and $H$, let $G overset{mathrm{rb}}{{longrightarrow}} H$ denote the property that for every proper edge colouring of $G$ there is a rainbow copy of $H$ in $G$. Extending a result of Nenadov, Person, v{S}kori{c} and Steger [J. Combin. T
A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses at most $k$ colors. For an integer $kgeq 1$, the Gallai-Ramsey number $GR_k(H)$ of a given graph $H$
We prove that every family of (not necessarily distinct) odd cycles $O_1, dots, O_{2lceil n/2 rceil-1}$ in the complete graph $K_n$ on $n$ vertices has a rainbow odd cycle (that is, a set of edges from distinct $O_i$s, forming an odd cycle). As part
If $G$ is a graph and $vec H$ is an oriented graph, we write $Gto vec H$ to say that every orientation of the edges of $G$ contains $vec H$ as a subdigraph. We consider the case in which $G=G(n,p)$, the binomial random graph. We determine the thresho