Related papers: Directed graphs with lower orientation Ramsey thre…

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…

An ordered hypergraph is a hypergraph $H$ with a specified linear ordering of the vertices, and the appearance of an ordered hypergraph $G$ in $H$ must respect the specified order on $V(G)$. In on-line Ramsey theory, Builder iteratively…

A random graph order is a partial order obtained from a random graph on $[n]$ by taking the transitive closure of the adjacency relation. The dimension of the random graph orders from random bipartite graphs $B(n,n,p)$ and from $G(n,p)$…

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula…

For given graphs $G_{1}, G_{2}$ and $G$, let $G\rightarrow (G_{1}, G_{2})$ denote that each red-blue-coloring of $E(G)$ yields a red copy of $G_{1}$ or a blue copy of $G_{2}$. Arag{\~a}o, Marciano and Mendon{\c c}a [L. Arag{\~a}o, J. Pedro…

For given graphs $G$ and $H,$ the \emph{Ramsey number} $R(G,H)$ is the least natural number $n$ such that for every graph $F$ of order $n$ the following condition holds: either $F$ contains $G$ or the complement of $F$ contains $H.$ In this…

For simple graphs $G$ and $H$, their size Ramsey number $\hat{r}(G,H)$ is the smallest possible size of $F$ such that for any red-blue coloring of its edges, $F$ contains either a red $G$ or a blue $H$. Similarly, we can define the…

We study the maximum dimension $d=d(n,p)$ for which an Erd\H{o}s-R\'enyi $G(n,p)$ random graph is $d$-rigid. Our main results reveal two different regimes of rigidity in $G(n,p)$ separated at $p_c=C_*\log n/n,~C_*=2/(1-\log 2)$ -- the point…

One of the central questions in Ramsey theory asks how small can be the size of the largest clique and independent set in a graph on $N$ vertices. By the celebrated result of Erd\H{o}s from 1947, the random graph on $N$ vertices with edge…

Let $H$ be a graph on $h$ vertices. The number of induced copies of $H$ in a graph $G$ is denoted by $i_H(G)$. Let $i_H(n)$ denote the maximum of $i_H(G)$ taken over all graphs $G$ with $n$ vertices. Let $f(n,h) = \Pi_{i}^h a_i$ where…

Let $R(H_1,H_2)$ denote the Ramsey number for the graphs $H_1, H_2$, and let $J_k$ be $K_k{-}e$. We present algorithms which enumerate all circulant and block-circulant Ramsey graphs for different types of graphs, thereby obtaining several…

We investigate the genus $g(n,m)$ of the Erd\H{o}s-R\'enyi random graph $G(n,m)$, providing a thorough description of how this relates to the function $m=m(n)$, and finding that there is different behaviour depending on which `region' $m$…

We consider $p$-orientations, which are defined to be orientations of $d$-regular graphs such that every vertex either has in-degree $p$ or out-degree $p$. These generalise the orientations considered in Jaeger's conjecture, where $d=4p+1$.…

Let $\mathcal{F}$ be a collection of $r$-uniform hypergraphs, and let $0 < p < 1$. It is known that there exists $c = c(p,\mathcal{F})$ such that the probability of a random $r$-graph in $G(n,p)$ not containing an induced subgraph from…

For random graphs, the containment problem considers the probability that a binomial random graph $G(n,p)$ contains a given graph as a substructure. When asking for the graph as a topological minor, i.e., for a copy of a subdivision of the…

We determine the sharp threshold for Hamilton cycles in randomly perturbed sparse graphs. For any $\alpha=\alpha(n)=o(1)$, let $G_{\alpha}$ be an $n$-vertex graph with minimum degree $\delta(G_{\alpha})\ge\alpha n$. We prove that if…

We study the random graph G_{n,\lambda/n} conditioned on the event that all vertex degrees lie in some given subset S of the non-negative integers. Subject to a certain hypothesis on S, the empirical distribution of the vertex degrees is…

The upper tail problem in the Erd\H{o}s--R\'enyi random graph $G\sim\mathcal{G}_{n,p}$ asks to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor $1+\delta$. Chatterjee and Dembo…

In this paper, we consider the question of computing sparse subgraphs for any input directed graph $G=(V,E)$ on $n$ vertices and $m$ edges, that preserves reachability and/or strong connectivity structures. We show $O(n+\min\{|{\cal…

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…