Offline thresholds for Ramsey-type games on random graphs

Ramsey games with giants

The classical result in the theory of random graphs, proved by Erdős and Rényi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey flavor. Now, each random edge that arrives in the sequence of rounds must be colored with one of r colors. The goal can be either to create a giant component in e...

On Balanced Coloring Games in Random Graphs

Consider the balanced Ramsey game, in which a player has r colors and where in each round r random edges of an initially empty graph on n vertices are presented. The player has to immediately assign a different color to each edge and her goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. The Achlioptas game is similar, but the player only loses when sh...

Long cycles in locally expanding graphs, with applications

We provide sufficient conditions for the existence of long cycles in locally expanding graphs, and present applications of our conditions and techniques to Ramsey theory, random graphs and positional games.

Online Ramsey Games in Random Graphs

Consider the following one-player game on a graph with n vertices. The edges are presented one by one to the player in a random order. One of r available colors has to be assigned to each edge immediately. The player’s objective is to color as many edges as possible without creating a monochromatic copy of some fixed graph F . We prove a lower bound of n on the typical duration of this game, wh...

Sharp thresholds for certain Ramsey properties of random graphs