Related papers: Ramsey upper density of infinite graph factors

For a given graph $H$, the Ramsey number $r(H)$ is the minimum $N$ such that any 2-edge-coloring of the complete graph $K_N$ yields a monochromatic copy of $H$. Given a positive integer $n$, a \emph{fan }$F_n$ is a graph formed by $n$…

A finite Euclidean set is diameter-Ramsey if, for every number of colors, some finite same-diameter witness has the property that every coloring of the witness contains a monochromatic congruent copy of the set. Frankl, Pach, Reiher and…

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…

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and…

The induced Ramsey number $r_{\mathrm{ind}}(F)$ of a $k$-uniform hypergraph $F$ is the smallest natural number $n$ for which there exists a $k$-uniform hypergraph $G$ on $n$ vertices such that every two-coloring of the edges of $G$ contains…

We determine the asymptotic behavior of the maximum subgraph density of large random graphs with a prescribed degree sequence. The result applies in particular to the Erd\H{o}s-R\'{e}nyi model, where it settles a conjecture of Hajek [IEEE…

Given a graph $G$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ if $V(G)\subset V(\mathcal{H})$ and there is a bijection $f:E(G)\rightarrow E(\mathcal{H})$ such that for any edge $e$ of $G$ we have $e\subset f(e)$. We study Ramsey…

The Ramsey multiplicity problem asks for the minimum asymptotic density of monochromatic labelled copies of a graph $H$ in a red/blue colouring of the edges of $K_n$. We introduce an off-diagonal generalization in which the goal is to…

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…

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…

The universal homogeneous triangle-free graph, constructed by Henson and denoted $\mathcal{H}_3$, is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with…

The anti-Ramsey number $\mathrm{ar}(n,F)$ of an $r$-graph $F$ is the minimum number of colors needed to color the complete $n$-vertex $r$-graph to ensure the existence of a rainbow copy of $F$. We establish a removal-type result for the…

Given a sequence $\mathbf{k} := (k_1,\ldots,k_s)$ of natural numbers and a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges…

We introduce a new method for computing bounds on the independence number and fractional chromatic number of classes of graphs with local constraints, and apply this method in various scenarios. We establish a formula that generates a…

Let $\Delta_s=R(K_3,K_s)-R(K_3,K_{s-1})$, where $R(G,H)$ is the Ramsey number of graphs $G$ and $H$ defined as the smallest $n$ such that any edge coloring of $K_n$ with two colors contains $G$ in the first color or $H$ in the second color.…

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…

For any graph $F$ and any integer $r\geq 2$, the \emph{online vertex-Ramsey density of $F$ and $r$}, denoted $m^*(F,r)$, is a parameter defined via a deterministic two-player Ramsey-type game (Painter vs.\ Builder). This parameter was…

We investigate Ramsey numbers of bounded degree graphs and provide an interpolation between known results on the Ramsey numbers of general bounded degree graphs and bounded degree graphs of small bandwidth. Our main theorem implies that…

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…

The set-colouring Ramsey number $R_{r,s}(k)$ is defined to be the minimum $n$ such that if each edge of the complete graph $K_n$ is assigned a set of $s$ colours from $\{1,\ldots,r\}$, then one of the colours contains a monochromatic clique…