Related papers: Multipartite Ramsey number $m_j(K_m, nK_2)$

For graphs $G_0$, $G_1$ and $G_2$, write $G_0\longmapsto(G_1, G_2)$ if each red-blue-edge-coloring of $G_0$ yields a red $G_1$ or a blue $G_2$. The Ramsey number $r(G_1, G_2)$ is the minimum number $n$ such that the complete graph…

Given two graphs $G$ and $H$, the {Ramsey number} $R(G,H)$ is the smallest positive integer $N$ such that every 2-coloring of the edges of $K_{N}$ contains either a red $G$ or a blue $H$. Let $K_{N-1}\sqcup K_{1,k}$ be the graph obtained…

It is well-known that in every $r$-coloring of the edges of the complete bipartite graph $K_{n,n}$ there is a monochromatic connected component with at least ${2n\over r}$ vertices. It would be interesting to know whether we can…

Given two graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any coloring of the edges of $K_N$ in red or blue yields a red $G$ or a blue $H$. Let $v(G)$ be the number of vertices of $G$ and $\chi(G)$ be the…

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…

Given a graph $H$, the Ramsey number $R(H)$ is the smallest positive integer $n$ such that every $2$-edge-colouring of $K_n$ yields a monochromatic copy of $H$. We write $mH$ to denote the union of $m$ vertex-disjoint copies of $H$. The…

The ordered Ramsey number of a graph $G^<$ with a linearly ordered vertex set is the smallest positive integer $N$ such that any two-coloring of the edges of the ordered complete graph on $N$ vertices contains a monochromatic copy of $G^<$…

We show that for any $2$-local colouring of the edges of the balanced complete bipartite graph $K_{n,n}$, its vertices can be covered with at most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or…

The Ramsey number $\mathrm{R}(G_1,G_2)$ is the smallest integer $N$ such that any red-blue coloring of the edges of the complete graph $K_N$ contains either a red copy of $G_1$ or a blue copy of $G_2$. In 2022, the third author and others…

Gallai-colorings are edge-colored complete graphs in which there are no rainbow triangles. Within such colored complete graphs, we consider Ramsey-type questions, looking for specified monochromatic graphs. In this work, we consider…

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula…

Let $G, H$ be two non-empty graphs and $k$ be a positive integer. The Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ is defined as the minimum positive integer $N$ such that for all $n\geq N$, every $k$-edge-coloring of $K_n$ contains…

For given simple graphs $G_1, G_2, \ldots , G_t$, the Ramsey number $R(G_1, G_2, \ldots, G_t)$ is the smallest positive integer $n$ such that if the edges of the complete graph $K_n$ are partitioned into $t$ disjoint color classes giving…

For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest integer $N$ such that every graph $\Gamma$ on $N$ vertices contains $G$ or its complement $\overline{\Gamma}$ contains $H$ as a subgraph. In graph Ramsey theory, the…

For a graph G=(V,E), a hypergraph H is called Berge-G if there is a bijection f from E(G) to E(H) such that for each e in E(G), e is a subset of f(e). The set of all Berge-G hypergraphs is denoted B(G). For integers k>1, r>1, and a graph G,…

For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is guaranteed…

For two given graphs $G_1$ and $G_2$, the Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\bar{G}$ contains a $G_2$. In this note, we determined the Ramsey…

The Ramsey number r(K_3,Q_n) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K_N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erd\H{o}s…

The Ramsey number $r(t;\ell)$ is the smallest $n$ such that every $\ell$-coloring of the edges of $K_n$ gives a monochromatic $K_{t}$. In recent years, there have been several improvements on asymptotic lower bounds for these numbers when…

Given two graphs $G$ and $H$, the \emph{general $k$-colored Gallai-Ramsey number} $\operatorname{gr}_k(G:H)$ is defined to be the minimum integer $m$ such that every $k$-coloring of the complete graph on $m$ vertices contains either a…