Balanced online Ramsey games in random graphs

Consider the following one-player game. Starting with the empty graph on n vertices, in every step r new edges are drawn uniformly at random and inserted into the current graph. These edges have to be colored immediately with r available colors, subject to the restriction that each color is used for exactly one of these edges. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove explicit threshold functions for the duration of this game for an arbitrary number of colors r and a large class of graphs F. This extends earlier work for the case r = 2 by Marciniszyn, Mitsche, and Stojakovic. We also prove a similar threshold result for the vertex-coloring analogue of this game.