Online Ramsey games for triangles in random graphs

Abstract

In the online F -avoidance edge-coloring game with r colors, a graph on n vertices is generated by randomly adding a new edge at each stage. The player must color each new edge as it appears; the goal is to avoid a monochromatic copy of F . Let N 0 ( F , r , n ) be the threshold function for the number of edges that the player is asymptotically almost surely able to paint before he/she loses. Even when F = K 3 , the order of magnitude of N 0 ( 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 .