Online Ramsey games for triangles in random graphs

13 years 4 months ago

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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, Sp

József Balogh, Jane Butterfield

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