Related papers: Ordered Size Ramsey Number of Paths
An oriented graph is a directed graph with no bi-directed edges, i.e. if $xy$ is an edge then $yx$ is not an edge. The oriented size Ramsey number of an oriented graph $H$, denoted by $r(H)$, is the minimum $m$ for which there exists an…
An ordered graph is a graph whose vertex set is equipped with a total order. The ordered complete graph $K_N^<$ is the complete graph with vertex set $[N]$ equipped with the natural ordering of the integers. Given an ordered graph $H$, the…
An ordered graph $H$ on $n$ vertices is a graph whose vertices have been labeled bijectively with $\{1,...,n\}$. The ordered Ramsey number $r_<(H)$ is the minimum $n$ such that every two-coloring of the edges of the complete graph $K_n$…
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…
Given a labeled graph $H$ with vertex set $\{1, 2,\ldots,n\}$, the ordered Ramsey number $r_<(H)$ is the minimum $N$ such that every two-coloring of the edges of the complete graph on $\{1, 2, \ldots,N\}$ contains a copy of $H$ with…
Given two vertex-ordered graphs $G$ and $H$, the ordered Ramsey number $R_<(G,H)$ is the smallest $N$ such that whenever the edges of a vertex-ordered complete graph $K_N$ are red/blue-coloured, then there is a red (ordered) copy of $G$ or…
An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph…
Given a graph $H$, the size Ramsey number $r_e(H,q)$ is the minimal number $m$ for which there is a graph $G$ with $m$ edges such that every $q$-coloring of $G$ contains a monochromatic copy of $H$. We study the size Ramsey number of the…
Given a vertex-ordered graph $G$, the ordered Ramsey number $r_<(G)$ is the minimum integer $N$ such that every $2$-coloring of the edges of the complete ordered graph $K_N$ contains a monochromatic ordered copy of $G$. Motivated by a…
An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the…
The ordered Ramsey number of a graph $G^<$ with a linearly ordered vertex set is the smallest positive integer $N$ such that any two-coloring of the edges of the ordered complete graph on $N$ vertices contains a monochromatic copy of $G^<$…
For a $k$-uniform hypergraph $G$ with vertex set $\{1,\ldots,n\}$, the ordered Ramsey number $\operatorname{OR}_t(G)$ is the least integer $N$ such that every $t$-coloring of the edges of the complete $k$-uniform graph on vertex set…
An edge-ordered graph is a graph with a linear ordering of its edges. Two edge-ordered graphs are equivalent if their is an isomorphism between them preserving the ordering of the edges. The edge-ordered Ramsey number $r_{edge}(H; q)$ of an…
An ordered graph $G$ is a graph together with a specified linear ordering on the vertices, and its interval chromatic number is the minimum number of independent sets consisting of consecutive vertices that are needed to partition the…
A $k$-ordering of a graph $G$ assigns distinct order-labels from the set $\{1,\ldots,|G|\}$ to $k$ vertices in $G$. Given a $k$-ordering $H$, the ordered Ramsey number $R_<(H)$ is the minimum $n$ such that every edge-2-coloring of the…
An ordered graph is a graph with a linear ordering on its vertices. The online Ramsey game for ordered graphs $G$ and $H$ is played on an infinite sequence of vertices; on each turn, Builder draws an edge between two vertices, and Painter…
The size-Ramsey number $\hat{R}(F,r)$ 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 $r$ colours yields a monochromatic copy of $F$.…
An ordered graph is a graph equipped with a linear ordering of its vertex set. A pair of ordered graphs is Ramsey finite if it has only finitely many minimal ordered Ramsey graphs and Ramsey infinite otherwise. Here an ordered graph F is an…
An ordered graph $\mathcal{G}$ is a simple graph together with a total ordering on its vertices. The (2-color) Ramsey number of $\mathcal{G}$ is the smallest integer $N$ such that every 2-coloring of the edges of the complete ordered graph…
Given a hypergraph $H$, the size-Ramsey number $\hat{r}_2(H)$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that in any colouring of the edges of $G$ with two colours there is a…