Related papers: Matching-star size Ramsey numbers under connectivi…

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…

The linked double star $S_c(n,m)$, where $n \geq m \geq 0$, is the graph consisting of the union of two stars $K_{1,n}$ and $K_{1,m}$ with a path on $c$ vertices joining the centers. Its ramsey number $r(S_c(n,m))$ is the smallest integer…

For graphs $G_1, G_2, G_3$, the three-color Ramsey number $R(G_1,$ $G_2, G_3)$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph of order $n$ with 3 colors, then it contains a monochromatic copy…

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…

In this paper we prove several results on Ramsey numbers $R(H,F)$ for a fixed graph $H$ and a large graph $F$, in particular for $F = K_n$. These results extend earlier work of Erd\H{o}s, Faudree, Rousseau and Schelp and of Balister, Schelp…

For given graphs G1 and G2 the Ramsey number R(G1,G2), is the smallest positive integer n such that each blue-red edge coloring of the complete graph Kn contains a blue copy of G1 or a red copy of G2. In 1983, Erdos conjectured that there…

The Ramsey number $r(G)$ of a graph $G$ is the smallest integer $n$ such that any $2$ colouring of the edges of a clique on $n$ vertices contains a monochromatic copy of $G$. Determining the Ramsey number of $G$ is a central problem of…

Given positive integers $k$ and $\ell$ we write $G \rightarrow (K_k,K_\ell)$ if every 2-colouring of the edges of $G$ yields a red copy of $K_k$ or a blue copy of $K_\ell$ and we denote by $R(k)$ the minimum $n$ such that $K_n\rightarrow…

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 two given graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the smallest integer $N$ such that any red-blue edge-coloring of the complete graph $K_N$ contains a red $F$ or a blue $H$. When $F=H$, we simply write $R_2(H)$. For an…

Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Tur\'an number of H, RT_t(n, H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G where f(n) is larger than the maximum number of vertices in a…

Let $r(G,H)$ be the smallest integer $N$ such that for any $2$-coloring (say, red and blue) of the edges of $K\_n$, $n\geqslant N$, there is either a red copy of $G$ or a blue copy of $H$. Let $K\_n-K\_{1,s}$ be the complete graph on $n$…

A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy of H. Two graphs H and H' are Ramsey-equivalent if every graph G is Ramsey for H if and only if it is Ramsey for H'. In this paper, we study…

The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have, such that for any red/blue colouring of $G$, there is a monochromatic copy of $H$ in $G$. Recently, Conlon, Nenadov and Truji\'c…

The size-Ramsey number $\hat{R}(F)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours yields a monochromatic copy of $F$. In…

For fixed finite graphs $G$, $H$, a common problem in Ramsey theory is to study graphs $F$ such that $F \to (G,H)$, i.e. every red-blue coloring of the edges of $F$ produces either a red $G$ or a blue $H$. We generalize this study to…

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 $G$, $H$ and $K$ represent three graphs without loops or parallel edges and $n$ represent an integer. Given any red blue coloring of the edges of $G$, we say that $K \rightarrow (G,H)$, if there exists red copy of $G$ in $K$ or a blue…

We study two problems in graph Ramsey theory. In the early 1970's, Erd\H{o}s and O'Neil considered a generalization of Ramsey numbers. Given integers $n,k,s$ and $t$ with $n \ge k \ge s,t \ge 2$, they asked for the least integer…

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,…