Related papers: Gallai-Ramsey multiplicity for rainbow small trees

Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a…

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 $k$ colors. Given an integer $k\ge1$ and graphs $H_1, \ldots, H_k$, the Gallai-Ramsey…

A Gallai $k$-coloring is a $k$-edge coloring of a complete graph in which there are no rainbow triangles. For given graphs $G_1, G_2, G_3$ and nonnegative integers $r, s, t$ with that $k=r+s+t$, the $k$-colored Gallai-Ramsey number…

For graphs $G_1,\ldots,G_k$, the Ramsey number $R(G_1,\ldots,G_k)$ is the smallest positive integer $N$ such that every $k$-edge-coloring of $K_N$ contains a monochromatic copy of $G_i$ in color $i$ for some $i\in[k]$. The Gallai--Ramsey…

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…

The size-Ramsey number of a graph $G$ is the minimum number of edges in a graph $H$ such that every 2-edge-coloring of $H$ yields a monochromatic copy of $G$. Size-Ramsey numbers of graphs have been studied for almost 40 years with…

Given a graph $H$ and a positive integer $k$, the {\it $k$-colored Ramsey number} $R_k(H)$ is the minimum integer $n$ such that in every $k$-edge-coloring of the complete graph $K_{n}$, there is a monochromatic copy of $H$. Given two graphs…

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a…

The Ramsey multiplicity constant of a graph $H$ is the minimum proportion of copies of $H$ in the complete graph which are monochromatic under an edge-coloring of $K_n$ as $n$ goes to infinity. Graphs for which this minimum is…

For given simple graphs $H_1,H_2,\dots,H_c$, the multicolor Ramsey number $R(H_1,H_2,\dots,H_c)$ is defined as the smallest positive integer $n$ such that for an arbitrary edge-decomposition $\{G_i\}^c_{i=1}$ of the complete graph $K_n$, at…

For given graphs $G_{1}, G_{2}, ... , G_{k}, k \geq 2$, the multicolor Ramsey number $R(G_{1}, G_{2}, ... , G_{k})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph of order $n$ with $k$ colors,…

A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. Given a graph $H$ and an integer $k\geq 1$, the…

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 $k$ colors. We study Ramsey-type problems in Gallai colorings. Given an integer $k\ge1$ and…

Given a positive integer $s$, the $s$-colour size-Ramsey number of a graph $H$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that, in any colouring of $E(G)$ with $s$ colours, there is a…

Let H_1, ..., H_k be graphs. The multicolor Ramsey number r(H_1,...,H_k) is the minimum integer r such that in every edge-coloring of K_r by k colors, there is a monochromatic copy of H_i in color i for some 1 <= i <= k. In this paper, we…

We study Ramsey-type problems in Gallai-colorings. Given a graph $G$ and an integer $k\ge1$, the Gallai-Ramsey number $gr_k(K_3,G)$ is the least positive integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$…

Let $H\xrightarrow{s} G$ denote that any edge-coloring of $H$ by $s$ colors contains a monochromatic $G$. The degree Ramsey number $r_{\Delta}(G;s)$ is defined to be $\min\{\Delta(H):H\xrightarrow{s} G\}$, and the degree bipartite Ramsey…

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. Given an integer $k\ge1$ and graphs $H_1, \ldots, H_k$, the…

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…

A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. Let $G$ and $H$ be two graphs. The anti-Ramsey number $\ar(G, H)$ is the maximum number of colors of an edge-coloring of $G$ that does not contain a…