Related papers: Graph classes with linear Ramsey numbers

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…

For graphs $F$, $G$, and $H$, we write $F \to (G,H)$ if every red-blue coloring of the edges of $F$ produces a red copy of $G$ or a blue copy of $H$. The graph $F$ is said to be $(G,H)$-minimal if it is subgraph-minimal with respect to this…

The generalized Ramsey number $R(H, K)$ is the smallest positive integer $n$ such that for any graph $G$ with $n$ vertices either $G$ contains $H$ as a subgraph or its complement $\overline{G}$ contains $K$ as a subgraph. Let $T_n$ be a…

We create a framework for hereditary graph classes $\mathcal{G}^\delta$ built on a two-dimensional grid of vertices and edge sets defined by a triple $\delta=\{\alpha,\beta,\gamma\}$ of objects that define edges between consecutive columns,…

A hereditary class H of graphs is $\chi$-bounded if there is a $\chi$-binding function f such that for every $G$ in $H$, $\chi(G)$ less than or equal to $f(\omega(G))$. Here we prove that if a graph $G$ is free of 1. {Chair; P$_4$+K$_1$} or…

We show that for every graph $H$, there is a hereditary weakly sparse graph class $\mathcal C_H$ of unbounded treewidth such that the $H$-free (i.e., excluding $H$ as an induced subgraph) graphs of $\mathcal C_H$ have bounded treewidth.…

The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. While this definition only asks for a single monochromatic copy of…

An ordered graph is a simple graph with an ordering on its vertices. Define the ordered path $P_n$ to be the monotone increasing path with $n$ edges. The ordered size Ramsey number $\tilde{r}(P_r,P_s)$ is the minimum number $m$ for which…

A system of linear equations with integer coefficients is partition regular over a subset S of the reals if, whenever S\{0} is finitely coloured, there is a solution to the system contained in one colour class. It has been known for some…

Let $G_1, G_2, ..., G_t$ be graphs. The multicolor Ramsey number $R(G_1, G_2, ..., G_t)$ is the smallest positive integer $n$ such that if the edges of complete graph $K_n$ are partitioned into $t$ disjoint color classes giving $t$ graphs…

Let $G_n$ be a random geometric graph, and then for $q,p \in [0,1)$ we construct a "$(q,p)$-perturbed noisy random geometric graph" $G_n^{q,p}$ where each existing edge in $G_n$ is removed with probability $q$, while and each non-existent…

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

An ordered graph $\mathcal{G}$ is a simple graph together with a total ordering on its vertices. The (2-color) Ramsey number of $\mathcal{G}$ is the smallest integer $N$ such that every 2-coloring of the edges of the complete ordered graph…

In this paper, we establish that the class of $\{P_6, (2,2)\text{-broom}\}$-free graphs contains a subclass $\mathcal{L}_i$, defined by certain cutset conditions, whose chromatic number admits a linear $\chi$-bound. Building on recent…

Daligault, Rao and Thomass\'e asked whether every hereditary graph class that is well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev (JCTB 2017+) gave a negative answer to this question,…

The Ramsey number $r(G)$ of a graph $G$ is the minimum $N$ such that every red-blue coloring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of $G$. Determining or estimating these numbers is one of the…

The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $r$-uniform…

For graphs G and H, let the induced Ramsey number IR(H,G) be the smallest number of vertices in a graph F such that any coloring of the edges of F in red and blue, there is either a red induced copy of H or a blue induced copy of G. In this…

We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically…

The $r$-color size-Ramsey number of a graph $H$, denoted by $\widehat{R}_r(H)$, is the minimum number of edges in a graph $G$ having the property that every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H$. Krivelevich…