Online Ramsey games for triangles in random graphs

In the online F -avoidance edge-coloring game with r colors, a graph on n vertices is generated by at each stage randomly adding a new edge. The player must color each new edge as it appears; his goal is to avoid a monochromatic copy of F . Let N0(F, r, n) be the threshold function for the number of edges that the player is asymptotically almost surely able to paint before he loses. Even when F = K3, the order of magnitude of N0(F, r, n) is unknown for r ≥ 3. In particular, the only known upper bound is the threshold function for the number of edges in the offline version of the problem, in which an entire random graph on n vertices with m edges is presented to the player to be r edge-colored. We improve the upper bound for the online triangle-avoidance game with r colors, providing the first result that separates the online threshold function from the offline bound for r ≥ 3. This supports a conjecture of Marciniszyn, Spöhel, and Steger that the known lower bound is tight for cliques and cycles for all r.