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If $G$ is a graph and $vec H$ is an oriented graph, we write $Gto vec H$ to say that every orientation of the edges of $G$ contains $vec H$ as a subdigraph. We consider the case in which $G=G(n,p)$, the binomial random graph. We determine the threshold $p_{vec H}=p_{vec H}(n)$ for the property $G(n,p)to vec H$ for the cases in which $vec H$ is an acyclic orientation of a complete graph or of a cycle.
A graph $G$ is said to be $q$-Ramsey for a $q$-tuple of graphs $(H_1,ldots,H_q)$, denoted by $Gto_q(H_1,ldots,H_q)$, if every $q$-edge-coloring of $G$ contains a monochromatic copy of $H_i$ in color $i,$ for some $iin[q]$. Let $s_q(H_1,ldots,H_q)$ de
We prove that $s_r(K_k) = O(k^5 r^{5/2})$, where $s_r(K_k)$ is the Ramsey parameter introduced by Burr, ErdH{o}s and Lov{a}sz in 1976, which is defined as the smallest minimum degree of a graph $G$ such that any $r$-colouring of the edges of $G$ cont
The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic
The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. While this definition only asks for a single monochromatic copy of $H$, it is
For graphs $G$ and $H$, let $G overset{mathrm{rb}}{{longrightarrow}} H$ denote the property that for every proper edge colouring of $G$ there is a rainbow copy of $H$ in $G$. Extending a result of Nenadov, Person, v{S}kori{c} and Steger [J. Combin. T