Related papers: Constrained Ramsey Numbers

In Ramsey theory for graphs we are given a graph $G$ and we are required to find the least $n_0$ such that, for any $n\geq n_0$, any red/blue colouring of the edges of $K_n$ gives a subgraph $G$ all of whose edges are blue or all are red.…

For any countably infinite graph $G$, Ramsey's theorem guarantees an infinite monochromatic copy of $G$ in any $r$-coloring of the edges of the countably infinite complete graph $K_\mathbb{N}$. Taking this a step further, it is natural to…

An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can…

Let $R(G)$ be the two-colour Ramsey number of a graph $G$. In this note, we prove that for any non-decreasing function $n \leq f(n) \leq R(K_n)$, there exists a sequence of connected graphs $(G_n)_{n\in\mathbb N}$, with $|V(G_n)| = n$ for…

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 $n$…

The anti-Ramsey number, $AR(n,G)$, for a graph $G$ and an integer $n\geq|V(G)|$, is defined to be the minimal integer $r$ such that in any edge-colouring of $K_n$ by at least $r$ colours there is a multicoloured copy of $G$, namely, a copy…

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…

Given a finite family $\mathcal{F}$ of graphs, we say that a graph $G$ is "$\mathcal{F}$-free" if $G$ does not contain any graph in $\mathcal{F}$ as a subgraph. A vertex-colored graph $H$ is called "rainbow" if no two vertices of $H$ have…

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…

An $r$-uniform hypergraph ($r$-graph for short) is called linear if every pair of vertices belong to at most one edge. A linear $r$-graph is complete if every pair of vertices are in exactly one edge. The famous Brown-Erd\H{o}s-S\'os…

An ordered graph $H$ on $n$ vertices is a graph whose vertices have been labeled bijectively with $\{1,...,n\}$. The ordered Ramsey number $r_<(H)$ is the minimum $n$ such that every two-coloring of the edges of the complete graph $K_n$…

A graph $G$ is $q$-Ramsey for another graph $H$ if in any $q$-edge-colouring of $G$ there is a monochromatic copy of $H$, and the classic Ramsey problem asks for the minimum number of vertices in such a graph. This was broadened in the…

Given graphs $H_1, \dots, H_t$, a graph $G$ is $(H_1, \dots, H_t)$-Ramsey-minimal if every $t$-coloring of the edges of $G$ contains a monochromatic $H_i$ in color $i$ for some $i\in\{1, \dots, t\}$, but any proper subgraph of $G $ does not…

We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on $n$ vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges…

For graphs $F$ and $H$, let $f_{F,H}(n)$ be the minimum possible size of a maximum $F$-free induced subgraph in an $n$-vertex $H$-free graph. This notion generalizes the Ramsey function and the Erd\H{o}s--Rogers function. Establishing a…

We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively.…

For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed…

We say that a graph $G$ has the Ramsey property w.r.t.\ some graph $F$ and some integer $r\geq 2$, or $G$ is $(F,r)$-Ramsey for short, if any $r$-coloring of the edges of $G$ contains a monochromatic copy of $F$. R{\"o}dl and Ruci{\'n}ski…

An edge colored graph is said to contain rainbow-$F$ if $F$ is a subgraph and every edge receives a different color. In 2007, Keevash, Mubayi, Sudakov, and Verstra\"ete introduced the \emph{rainbow extremal number} $\mathrm{ex}^*(n,F)$, a…

The well-known Ramsey number $r(t,u)$ is the smallest integer $n$ such that every $K_t$-free graph of order $n$ contains an independent set of size $u$. In other words, it contains a subset of $u$ vertices with no $K_2$. Erd{\H o}s and…