相关论文: Two types of size Ramsey numbers for matchings of …

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

Given two graphs $G_1$ and $G_2$, the Ramsey number $r(G_1,G_2)$ refers to the smallest positive integer $N$ such that any graph $G$ with $N$ vertices contains $G_1$ as a subgraph, or the complement of $G$ contains $G_2$ as a subgraph. A…

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the…

Given graphs $F,H$ and $G$, we say that $G$ is $(F,H)_v$-Ramsey if every red/blue vertex colouring of $G$ contains a red copy of $F$ or a blue copy of $H$. Results of {\L}uczak, Ruci\'nski and Voigt and, subsequently, Kreuter determine the…

Let $r,\ell\geq2$ be integers. Given $r$-graphs $G$ and $F_1,\dots,F_\ell$, we write $G\to(F_1,\dots,F_\ell)$ if every $\ell$-edge-coloring of $G$ yields a monochromatic copy of $F_i$ in the $i$th color for some $1\leq i\leq\ell$, otherwise…

An ordered graph is a graph whose vertex set is equipped with a total order. The ordered complete graph $K_N^<$ is the complete graph with vertex set $[N]$ equipped with the natural ordering of the integers. Given an ordered graph $H$, the…

Often regarded as the study of how order emerges from randomness, Ramsey theory has played an important role in mathematics and computer science, giving rise to applications in numerous domains such as logic, parallel processing, and number…

Given any graph $H$, a graph $G$ is said to be $q$-Ramsey for $H$ if every coloring of the edges of $G$ with $q$ colors yields a monochromatic subgraph isomorphic to $H$. Further, such a graph $G$ is said to be minimal $q$-Ramsey for $H$ if…

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 $…

A graph $G$ is said to be $q$-Ramsey for a $q$-tuple of graphs $(H_1,\ldots,H_q)$, denoted by $G\to_q(H_1,\ldots,H_q)$, if every $q$-edge-coloring of $G$ contains a monochromatic copy of $H_i$ in color $i,$ for some $i\in[q]$. Let…

In this paper, we study Ramsey-type problems for directed graphs. We first consider the $k$-colour oriented Ramsey number of $H$, denoted by $\overrightarrow{R}(H,k)$, which is the least $n$ for which every $k$-edge-coloured tournament on…

An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that…

For every $k\ge 2$ and $\Delta$, we prove that there exists a constant $C_{\Delta,k}$ such that the following holds. For every graph $H$ with $\chi(H)=k$ and every tree with at least $C_{\Delta,k}|H|$ vertices and maximum degree at most…

The hypergraph Ramsey number of two $3$-uniform hypergraphs $G$ and $H$, denoted by $R(G,H)$, is the least integer $N$ such that every red-blue edge-coloring of the complete $3$-uniform hypergraph on $N$ vertices contains a red copy of $G$…

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 $G_2$. We denote by $P_n$ the path on…

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…

For an integer $r\geq 2$ and bipartite graphs $H_i$, where $1\leq i\leq r$, the bipartite Ramsey number $br(H_1,H_2,\ldots,H_r)$ is the minimum integer $N$ such that any $r$-edge coloring of the complete bipartite graph $K_{N,N}$ contains a…

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…

In this paper, we determine the $r$-colour size Ramsey number of the path $P_k$, up to constants. In particular, for every fixed $r \geq 2$ and $k \geq 100\log r$, we have \[ \widehat{R}_r(P_k)=\Theta((r^2 \log r) \, k).\] Perhaps…

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph isomorphic to S or a rainbow (all…