Related papers: On the size-Ramsey number of hypergraphs
Let the grid graph $G_{M\times N}$ denote the Cartesian product $K_M \square K_N$. For a fixed subgraph $H$ of a grid, we study the off-diagonal Ramsey number $\operatorname{gr}(H, K_k)$, which is the smallest $N$ such that any red/blue…
We investigate Ramsey properties of a random graph model in which random edges are added to a given dense graph. Specifically, we determine lower and upper bounds on the function $p=p(n)$ that ensures that for any dense graph $G_n$ a.a.s.…
Ramsey's theorem, concerning the guarantee of certain monochromatic patterns in large enough edge-coloured complete graphs, is a fundamental result in combinatorial mathematics. In this work, we highlight the connection between this…
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 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…
We say that a graph $G$ is anti-Ramsey for a graph $H$ if any proper edge-colouring of $G$ yields a rainbow copy of $H$, i.e. a copy of $H$ whose edges all receive different colours. In this work we determine the threshold at which the…
The multicolor Ramsey number problem asks, for each pair of natural numbers $\ell$ and $t$, for the largest $\ell$-coloring of a complete graph with no monochromatic clique of size $t$. Recent works of Conlon-Ferber and Wigderson have…
An edge-colored graph is called \textit{rainbow graph} if all the colors on its edges are distinct. For a given positive integer $n$ and a family of graphs $\mathcal{G}$, the anti-Ramsey number $ar(n, \mathcal{G})$ is the smallest number of…
We call a subgraph of an edge-colored graph rainbow subgraph, if all of its edges have different colors. The anti-Ramsey number of a graph $G$ in a complete graph $K_{n}$, denoted by $ar(K_{n}, G)$, is the maximum number of colors in an…
Gallai-colorings are edge-colored complete graphs in which there are no rainbow triangles. Within such colored complete graphs, we consider Ramsey-type questions, looking for specified monochromatic graphs. In this work, we consider…
For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, where…
For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $r$ such that any red-blue edge coloring of $K_r$ contains a red $G$ or a blue $H$. The path-critical Ramsey number $R_{\pi}(G,H)$ is the largest $n$ such that any red-blue…
Given graphs $G$, $H_1$, and $H_2$, let $G\xrightarrow{\text{mr}}(H_1,H_2)$ denote the property that in every edge colouring of $G$ there is a monochromatic copy of $H_1$ or a rainbow copy of $H_2$. The constrained Ramsey number, defined as…
Determine the size of $r$-graphs with given graph parameters is an interesting problem. Chv\'atal and Hanson (JCTB, 1976) gave a tight upper bound of the size of 2-graphs with restricted maximum degree and matching number; Khare (DM, 2014)…
For a given graph $H$, the anti-Ramsey number of $H$ is the maximum number of colors in an edge-coloring of a complete graph which does not contain a rainbow copy of $H$. In this paper, we extend the decomposition family of graphs to 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…
Let $R_h(k; \ell)$ be the smallest integer $n$ such that any edge coloring of a complete graph on $n$ vertices in $\ell$ colors results in a monochromatic $K_k$-minor, in other words, a graph with Hadwiger number $k$, i.e., a graph that…
In this paper, we address problems related to parameters concerning edge mappings of graphs. Inspired by Ramsey's Theorem, the quantity $m(G, H)$ is defined to be the minimum number $n$ such that for every $f: E(K_n) \rightarrow E(K_n)$…
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…
The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family F (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete…