Online Ramsey Games in Random Graphs

Abstract

Consider the following one-player game. Starting with the empty graph onnvertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one ofravailable colours. The player's goal is to avoid creating a monochromatic copy of some fixed graphFfor as long as possible. We prove a lower bound ofnβ(F,r)on the typical duration of this game, where β(F,r) is a function that is strictly increasing inrand satisfies limr→∞β(F,r) = 2 − 1/m2(F), wheren2−1/m2(F)is the threshold of the corresponding offline colouring problem.