Ramsey Games Against a One-Armed Bandit

We study the following one-person game against a random graph: the Player’s goal is to 2-colour a random sequence of edges e1, e2, . . . of a complete graph on n vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochromatic triangle is created. The online version of the game requires that the Player should colour each edge when it comes, before looking at the next edge. While it is not hard to prove that the expected length of this game is about n, the proof of the upper bound suggests the following relaxation: instead of colouring online, the random graph is generated in only two rounds, and the Player colours the edges of the first round before the edges of the second round are generated. Given the size of the first round, how many edges can there be in the second round if the Player is to win with reasonable probability? In the extreme case, when the first round consists of a random graph with cn edges, where c is a positive constant, we show that the Player can win only if constantly many edges are generated in the second round. The analysis of the two-round version of the game is based on a delicate lemma concerning edge-coloured random graphs.