Related papers: On the anti-Ramsey threshold

Given a graph $H$, the maximal anti-Ramsey function $f(n,e,H)$ denotes the minimum integer $f$ for which there exists an $n$-vertex graph $G$ with at least $e$ edges admitting an edge-coloring with $f$ colors in which each copy of $H$ in…

In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve…

We study the Maker-Breaker $H$-game played on the edge set of the random graph $G_{n,p}$. In this game two players, Maker and Breaker, alternately claim unclaimed edges of $G_{n,p}$, until all the edges are claimed. Maker wins if he claims…

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 introduce and study a variant of Ramsey numbers for edge-ordered graphs, that is, graphs with linearly ordered sets of edges. The edge-ordered Ramsey number $\overline{R}_e(\mathfrak{G})$ of an edge-ordered graph $\mathfrak{G}$ is the…

We give asymptotically optimal constructions in generalized Ramsey theory using results about conflict-free hypergraph matchings. For example, we present an edge-coloring of $K_{n,n}$ with $2n/3 + o(n)$ colors such that each $4$-cycle…

For a graph G=(V,E), a hypergraph H is called Berge-G if there is a bijection f from E(G) to E(H) such that for each e in E(G), e is a subset of f(e). The set of all Berge-G hypergraphs is denoted B(G). For integers k>1, r>1, and a graph G,…

Let $H$ be an edge colored hypergraph. We say that $H$ contains a \emph{rainbow} copy of a hypergraph $S$ if it contains an isomorphic copy of $S$ with all edges of distinct colors. We consider the following setting. A randomly edge colored…

For a given graph $F$ we consider the family of (finite) graphs $G$ with the Ramsey property for $F$, that is the set of such graphs $G$ with the property that every two-colouring of the edges of $G$ yields a monochromatic copy of $F$. For…

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 prove the following: Fix an integer $k\geq 2$, and let $T$ be a real number with $T\geq 1.5$. Let $\cH=(V,\cE_2\cup \cE_3\cup\dots\cup\cE_k)$ be a non-uniform hypergraph with the vertex set $V$ and the set $\cE_i$ of edges of size…

Given graphs $G$ and $H$ and a positive integer $k$, the \emph{Gallai-Ramsey number}, denoted by $gr_{k}(G : H)$ is defined to be the minimum integer $n$ such that every coloring of $K_{n}$ using at most $k$ colors will contain either a…

R\"odl and Ruci\'nski (1990) established Ramsey's theorem for random graphs. In particular, for fixed integers $r$, $\ell\geq 2$ they showed that $\hat p_{K_\ell,r}(n)=n^{-\frac{2}{\ell+1}}$ is a threshold for the Ramsey property that every…

We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices. The Ramsey number r(H) of a…

For every $k\ge 2$ and $\Delta$, we prove that there exists a constant $C_{\Delta,k}$ such that the following holds. For every graph $H$ with $\chi(H)=k$ and every tree with at least $C_{\Delta,k}|H|$ vertices and maximum degree at most…

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…

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…

For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. An $R$-graph $\mathcal{H}$ is \emph{covering} if every vertex pair of $\mathcal{H}$…

The anti-Ramsey problem was introduced by Erd\H{o}s, Simonovits and S\'{o}s in 1970s. The anti-Ramsey number of a hypergraph $\mathcal{H}$, $ar(n,s, \mathcal{H})$, is the smallest integer $c$ such that in any coloring of the edges of the…

Motivated by anti-Ramsey numbers introduced by Erd\H{o}s, Simonovits and S\'os in 1975, we study the anti-Ramsey problem when host graphs are plane triangulations. Given a positive integer $n$ and a planar graph $H$, let $\mathcal{T}_n(H)$…