Related papers: On-line Ramsey numbers
Consider a game played on the edge set of the infinite clique by two players, Builder and Painter. In each round, Builder chooses an edge and Painter colours it red or blue. Builder wins by creating either a red copy of $G$ or a blue copy…
The $(m,n)$-online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge on each turn and Painter immediately paints it red or blue. Builder's…
Online Ramsey game is played between Builder and Painter on an infinite board $K_{\mathbb N}$. In every round Builder selects an edge, then Painter colors it red or blue. Both know target graphs $H_1$ and $H_2$. Builder aims to create…
In this paper we consider a game played on the edge set of the infinite clique $K_\mathbb{N}$ by two players, Builder and Painter. In each round of the game, Builder chooses an edge and Painter colors it red or blue. Builder wins when…
Given two graphs $G$ and $H$, the online Ramsey number $\tilde{r}(G,H)$ is defined to be the minimum number of rounds that Builder can always guarantee a win in the following $(G, H)$-online Ramsey game between Builder and Painter. Starting…
The online Ramsey number $\tilde r(G,H)$ is defined via a Builder--Painter game on an empty graph with countably many vertices. In each round, Builder reveals an edge, which Painter immediately colors either red or blue. Builder wins once a…
The online ordered Ramsey game is played between two players, Builder and Painter, on an infinite sequence of vertices with ordered graphs $(G_1,G_2)$, which have linear orderings on their vertices. On each turn, Builder first selects an…
The online Ramsey game for graphs $G$ and $H$ is played on the infinite complete graph $K_\mathbb{N}$. Each round, Builder chooses an edge, and Painter colors it red or blue. The online Ramsey number $\tilde{r}(G,H)$ is the smallest integer…
For two graphs $G_1$ and $G_2$, the online Ramsey number $\tilde{r}(G_1,G_2)$ is the smallest number of edges that Builder draws on an infinite empty graph to guarantee that there is either a red copy of $G_1$ or a blue copy of $G_2$, under…
Given two graphs $H_1$ and $H_2$, an online Ramsey game is played on the edge set of $K_\mathbb{N}$. In every round Builder selects an edge and Painter colors it red or blue. Builder is trying to force Painter to create a red copy of $H_1$…
The restricted $(m,n;N)$-online Ramsey game is a game played between two players, Builder and Painter. The game starts with $N$ isolated vertices. Each turn Builder picks an edge to build and Painter chooses whether that edge is red or…
Given two graphs $G$ and $H$, a size Ramsey game is played on the edge set of $K_\mathbb{N}$. In every round, Builder selects an edge and Painter colours it red or blue. Builder's goal is to force Painter to create a red copy of $G$ or a…
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…
An online Ramsey game is a game between Builder and Painter, alternating in turns. They are given a graph $H$ and a graph $G$ of an infinite set of independent vertices. In each round Builder draws an edge and Painter colors it either red…
Consider the following Ramsey game played on the edge set of $K_{\mathbb N}$. In every round, Builder selects an edge and Painter colours it red or blue. Builder's goal is to force Painter to create a red copy of a path $P_k$ on $k$…
The online Ramsey turnaround game is a game between two players, Builder and Painter, on a board of $n$ vertices using $3$ colors, for a fixed graph $H$ on at most $n$ vertices. The goal of Painter is to force a monochromatic copy of $H$,…
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…
Consider the following one-player game played on an initially empty graph with $n$ vertices. At each stage a randomly selected new edge is added and the player must immediately color the edge with one of $r$ available colors. Her objective…
Consider a two-player game between players Builder and Painter. Painter begins the game by picking a coloring of the edges of $K_n$, which is hidden from Builder. In each round, Builder points to an edge and Painter reveals its color.…
Given a class $\mathcal{C}$ of graphs and a fixed graph $H$, the online Ramsey game for $H$ on $\mathcal C$ is a game between two players Builder and Painter as follows: an unbounded set of vertices is given as an initial state, and on each…