相关论文: The anti-Ramsey threshold of complete graphs

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…

The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that for a given $k$-uniform hypergraph (or $k$-graph) $H$, if $n$ is sufficiently large then any colouring of the edges of the complete $k$-graph $K^{(k)}_n$ gives rise…

An edge-colored hypergraph is called \emph{a rainbow hypergraph} if all the colors on its edges are distinct. Given two positive integers $n,r$ and an $r$-uniform hypergraph $\mathcal{G}$, the anti-Ramsey number $ar_r(n,\mathcal{G})$ is…

In this work, we develop a unified framework for establishing sharp threshold results for various Ramsey properties. To achieve this, we view such properties as non-colourability of auxiliary hypergraphs. Our main technical result gives…

The size-Ramsey number $R^{(k)}(H)$ of a $k$-uniform hypergraph $H$ is the minimum number of edges in a $k$-uniform hypergraph $G$ with the property that every `$2$-edge coloring' of $G$ contains a monochromatic copy of $H$. For $k\ge2$ and…

We call an edge colouring of a graph G a rainbow colouring if every pair of vertices is joined by a rainbow path, i.e., a path where no two edges have the same colour. The minimum number of colours required for a rainbow colouring of the…

An edge-colored graph is called rainbow if all the colors on its edges are distinct. Given a positive integer n and a graph G, the anti-Ramsey number ar(n,G) is the maximum number of colors in an edge-coloring of K_{n} with no rainbow copy…

The square $G^2$ of a graph $G$ is the graph on $V(G)$ with a pair of vertices $uv$ an edge whenever $u$ and $v$ have distance $1$ or $2$ in $G$. Given graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum $N$ such that whenever 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 $\mathcal{T}_n(H)$…

The size-Ramsey number $\hat{R}(F)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours yields a monochromatic copy of $F$. In…

Let $G$ be a graph on $n$ vertices and let $k$ be a fixed positive integer. We denote by $\mathcal G_{\text{$k$-out}}(G)$ the probability space consisting of subgraphs of $G$ where each vertex $v\in V(G)$ randomly picks $k$ neighbors from…

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,…

Given two graphs $G$ and $H$, the \emph{general $k$-colored Gallai-Ramsey number} $\operatorname{gr}_k(G:H)$ is defined to be the minimum integer $m$ such that every $k$-coloring of the complete graph on $m$ vertices contains either a…

The canonical Ramsey theorem of Erd\H{o}s and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy…

We study a generalisation of the bipartite Ramsey numbers to blowups of graphs. For a graph $G$, denote the $t$-blowup of $G$ by $G[t]$. We say that $G$ is $r$-Ramsey for $H$, and write $G \stackrel{r}{\rightarrow} H$, if every…

Given two graphs $G$ and $H$, let $f(G,H)$ denote the maximum number $c$ for which there is a way to color the edges of $G$ with $c$ colors such that every subgraph $H$ of $G$ has at least two edges of the same color. Equivalently, any…

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…

We study edge-colorings of the complete $p$-graph on $n$ vertices that contain no three edges $A,B,C$ of distinct colors such that the symmetric difference of $A$ and $B$ is contained in $C$. For $p\ge3$ and $n\ge p+1$, we show that every…

Given a function $p : V(G)\to \mathbb N$ and an integer $k\ge 0$, define $p_k(G)$ as the number of vertices with $p(v)\ge k$. We say that $p_k(G)$ is bounded for all $\HH$-free graphs if there exists a constant $c=c(\HH)$ such that…

For a given graph $H$ and $n\geq 1$, let $f(n,H)$ denote the maximum number $c$ for which there is a way to color the edges of the complete graph $K_n$ with $c$ colors such that every subgraph $H$ of $K_n$ has at least two edges of the same…