Related papers: Minimal Ramsey graphs for cyclicity

The \emph{Ramsey multiplicity constant} of a graph $H$ is the limit as $n$ tends to infinity of the minimum density of monochromatic labeled copies of $H$ in a $2$-edge colouring of $K_n$. Fox and Wigderson recently identified a large…

Let $F$, $G$ and $H$ be simple graphs. We say $F \rightarrow (G, H)$ if for every $2$-coloring of the edges of $F$ there exists a monochromatic $G$ or $H$ in $F$. The Ramsey number $r(G, H)$ is defined as $r(G, H) = min\{|V (F)|: F…

The Ramsey multiplicity constant of a graph $H$ is the minimum proportion of copies of $H$ in the complete graph which are monochromatic under an edge-coloring of $K_n$ as $n$ goes to infinity. Graphs for which this minimum is…

For a finite family $\mathcal{F}$ of fixed graphs let $R_k(\mathcal{F})$ be the smallest integer $n$ for which every $k$-coloring of the edges of the complete graph $K_n$ yields a monochromatic copy of some $F\in\mathcal{F}$. We say that…

Bondy and Vince showed that every graph with minimum degree at least three contains two cycles of lengths differing by one or two.We prove the following average degree counterpart that every $n$-vertex graph $G$ with at least $\frac52(n-1)$…

For graphs $F$ and $G$, let $F\to G$ signify that any red/blue edge coloring of $F$ contains a monochromatic $G$. Denote by ${\cal G}(N,p)$ the random graph space of order $N$ and edge probability $p$. Using the regularity method, one can…

Motivated by an extremal problem on graph-codes that links coding theory and graph theory, Alon recently proposed a question aiming to find the smallest number $t$ such that there is an edge coloring of $K_{n}$ by $t$ colors with no copy of…

The grid Ramsey number $ G(r) $ is the smallest number $ n $ such that every edge-colouring of the grid graph $\Gamma_{n,n} := K_n \times K_n$ with $r$ colours induces a rectangle whose parallel edges receive the same colour. We show $ G(r)…

Bipartite Ramsey numbers is the smallest size of a complete bipartite graph $K_{N,N}$ such that every edge-coloring with a given number of colors inevitably yields a monochromatic copy of a prescribed bipartite graph. While exact values…

There has been extensive research on cycle lengths in graphs with large minimum degree. In this paper, we obtain several new and tight results in this area. Let $G$ be a graph with minimum degree at least $k+1$. We prove that if $G$ is…

The induced $q$-color size-Ramsey number $\hat{r}_{\text{ind}}(H;q)$ of a graph $H$ is the minimal number of edges a host graph $G$ can have so that every $q$-edge-coloring of $G$ contains a monochromatic copy of $H$ which is an induced…

We show that the minimum number of orientations of the edges of the n-vertex complete graph having the property that every triangle is made cyclic in at least one of them is $\lceil\log_2(n-1)\rceil$. More generally, we also determine the…

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)…

Ramsey's Theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdos conjectured that the random 2-edge-coloring minimizes the number of…

Anti-Ramsey theory was initiated in 1975 by Erd\H{o}s, Simonovits and S\'os, inspiring hundreds of publications since then. The present work is the third and last piece of our trilogy in which we introduce a far-reaching generalization via…

A graph $G$ is said to be Ramsey size-linear if $r(G,H) =O_G (e(H))$ for every graph $H$ with no isolated vertices. Erd\H{o}s, Faudree, Rousseau, and Schelp observed that $K_4$ is not Ramsey size-linear, but each of its proper subgraphs is,…

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…

In an $r$-coloring of edges of the complete graph on $n$ vertices, how many edges are there in the largest monochromatic connected component? A construction of Gy\'arf\'as shows that for infinitely many values of $r$, there exist colorings…

Let $k \in \mathbb{N}$ and let $H_1, H_2, \ldots, H_k$ be simple graphs such that for each $j \in \{ 1, 2, \ldots, k \}$, the vertex set of $H_j$ is $\{ 0, 1, 2, \ldots, n_j - 1 \}$ for some $n_j \in \mathbb{N}$. The ordered Ramsey number…

For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. Let the Ramsey number $R^r(BG,BG)$ be…