Related papers: Good Graph Hunting

Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If graph $G$ is connected, it is…

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…

Given simple graphs $H_{1},H_{2},\ldots,H_{c}$, the Ramsey number $r(H_{1},H_{2},\ldots,H_{c})$ is the smallest positive integer $n$ such that every edge-colored $K_{n}$ with $c$ colors contains a subgraph in color $i$ isomorphic to $H_{i}$…

Given graphs $G$ and $H$, we say that $G$ is $H$-$good$ if the Ramsey number $R(G,H)$ equals the trivial lower bound $(|G| - 1)(\chi(H) - 1) + \sigma(H)$, where $\chi(H)$ denotes the usual chromatic number of $H$, and $\sigma(H)$ denotes…

Given a pair of $k$-uniform hypergraphs $(G,H)$, the Ramsey number of $(G,H)$, denoted by $R(G,H)$, is the smallest integer $n$ such that in every red/blue-colouring of the edges of $K_n^{(k)}$ there exists a red copy of $G$ or a blue copy…

Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it…

For graph $G$, a connected graph $H$ of order $n$ is said to be $G$-good if $r(G,H)=(\chi(G)-1)(n-1)+s(G)$, where $\chi(G)$ is the chromatic number of $G$ and $s(G)$ is the minimum size of a color class in a $\chi(G)$-coloring of $G$. Let…

Given two graphs $G_1$ and $G_2$, the Ramsey number $R(G_1,G_2)$ denotes the smallest integer $N$ such that any red-blue coloring of the edges of $K_N$ contains either a red $G_1$ or a blue $G_2$. Let $G_1$ be a graph with chromatic number…

Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it…

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…

The Ramsey number $r(G,H)$ is the minimum $N$ such that every graph on $N$ vertices contains $G$ as a subgraph or its complement contains $H$ as a subgraph. For integers $n \geq k \geq 1$, the $k$-book $B_{k,n}$ is the graph on $n$ vertices…

Given two graphs $G$ and $H$, the $k$-colored Gallai-Ramsey number $gr_k(G : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the complete graph on $n$ vertices contains either a rainbow copy of $G$ or a…

The Ramsey number r(H) of a graph H is the smallest number n such that, in any two-colouring of the edges of K_n, there is a monochromatic copy of H. We study the Ramsey number of graphs H with t vertices and density \r, proving that r(H)…

Let $\mathcal{H}$ be a 3-uniform hypergraph. The multicolor Ramsey number $ r_k(\mathcal{H})$ is the smallest integer $n$ such that every coloring of $ \binom{[n]}{3}$ with $k$ colors has a monochromatic copy of $\mathcal{H}$. Let $…

Let $G_1, G_2, ..., G_t$ be graphs. The multicolor Ramsey number $R(G_1, G_2, ..., G_t)$ is the smallest positive integer $n$ such that if the edges of complete graph $K_n$ are partitioned into $t$ disjoint color classes giving $t$ graphs…

A graph $G$ is called $H$-good if $R(G,H)=(|G|-1)(\chi(H)-1)+\sigma(H)$, where $\sigma(H)$ denotes the size of the smallest color class in a $\chi(H)$-coloring of $H$. In Ramsey theory, it is an interesting problem to study whether a graph…

Let $G$ and $H$ be finite graphs. If for any two-coloring of the edges of a complete graph $K_n$, there is a copy of $G$ in the first color, red, or a copy of $H$ in the second color, blue, we will say $K_n\rightarrow (G,H)$. The Ramsey…

For two graphs $G^<$ and $H^<$ with linearly ordered vertex sets, the ordered Ramsey number $r_<(G^<,H^<)$ is the minimum $N$ such that every red-blue coloring of the edges of the ordered complete graph on $N$ vertices contains a red copy…

Given graphs $G$ and $H$ and a positive integer $k$, the \emph{Gallai-Ramsey number}, denoted by $gr_{k}(G : H)$ is defined to be the minimum integer $n$ such that every coloring of $K_{n}$ using at most $k$ colors will contain either 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…