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 Stojaković. We also prove a similar threshold result for the vertex-coloring analogue of this game. 1

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Abstract. Consider the balanced Ramsey game, in which a player has r colors and where in each step 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 she creates a copy of F in one distinguished color. We show that there is an infinite family of non-forests F for which the balanced Ramsey game has a different threshold than the Achlioptas game, settling an open question by Krivelevich et al. We also consider the natural vertex analogues of both games and show that their thresholds coincide for all graphs F, in contrast to our results for the edge case. 1.