Related papers: On vertex Ramsey graphs with forbidden subgraphs

We consider properties of edge-colored vertex-ordered graphs, i.e., graphs with a totally ordered vertex set and a finite set of possible edge colors. We show that any hereditary property of such graphs is strongly testable, i.e., testable…

A graph is "$H$-free" if it has no induced subgraph isomorphic to $H$. A conjecture of Conlon, Fox and Sudakov states that for every graph $H$, there exists $s>0$ such that in every $H$-free graph with $n>1$ vertices, either some vertex has…

A major line of research is discovering Ramsey-type theorems, which are results of the following form: given a graph parameter $\rho$, every graph $G$ with sufficiently large $\rho(G)$ contains a `well-structured' induced subgraph $H$ with…

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…

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…

Let $\mathcal{F}$ be a family of fixed graphs and let $d$ be large enough. For every $d$-regular graph $G$, we study the existence of a spanning $\mathcal{F}$-free subgraph of $G$ with large minimum degree. This problem is well-understood…

For a partially ordered set $(A, \le)$, let $G_A$ be the simple, undirected graph with vertex set $A$ such that two vertices $a \neq b\in A$ are adjacent if either $a \le b$ or $b \le a$. We call $G_A$ the \emph{partial order graph} or…

An important theme of recent research in Ramsey theory has been establishing pseudorandomness properties of Ramsey graphs. An $N$-vertex graph is called $C$-Ramsey if it has no homogeneous set of size $C\log N$. A theorem of Bukh and…

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…

Given a fixed integer $n$, we prove Ramsey-type theorems for the classes of all finite ordered $n$-colorable graphs, finite $n$-colorable graphs, finite ordered $n$-chromatic graphs, and finite $n$-chromatic graphs.

Consider the following random process: The vertices of a binomial random graph $G_{n,p}$ are revealed one by one, and at each step only the edges induced by the already revealed vertices are visible. Our goal is to assign to each vertex one…

We consider extremal edge-coloring problems inspired by the theory of anti-Ramsey / rainbow coloring, and further by odd-colorings and conflict-free colorings. Let $G$ be a graph, and $F$ any given family of graphs. For every integer $n…

The Ramsey number $R_X(p,q)$ for a class of graphs $X$ is the minimum $n$ such that every graph in $X$ with at least $n$ vertices has either a clique of size $p$ or an independent set of size $q$. We say that Ramsey numbers are linear in…

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…

We study Tur\'an and Ramsey-type problems on edge-colored graphs. An edge-colored graph is called {\em $\varepsilon$-balanced} if each color class contains at least an $\varepsilon$-proportion of its edges. Given a family $\mathcal{F}$ of…

The Ramsey number $R(F,H)$ is the minimum number $N$ such that any $N$-vertex graph either contains a copy of $F$ or its complement contains $H$. Burr in 1981 proved a pleasingly general result that for any graph $H$, provided $n$ is…

We investigate the impact of a high-degree vertex in Tur\'{a}n problems for degenerate hypergraphs (including graphs). We say an $r$-graph $F$ is bounded if there exist constants $\alpha, \beta>0$ such that for large $n$, every $n$-vertex…

A path in a vertex-colored graph is called \emph{vertex-rainbow} if its internal vertices have pairwise distinct colors. A graph $G$ is \emph{rainbow vertex-connected} if for any two distinct vertices of $G$, there is a vertex-rainbow path…

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…

A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it…