Probabilistic One-Player Ramsey Games via Deterministic Two-Player Games

Consider the following probabilistic one-player game: The board is a graph with n vertices, which initially contains no edges. In each step, a new edge is drawn uniformly at random from all non-edges and is presented to the player, henceforth called Painter. Painter must assign one of r available colors to each edge immediately, where r ≥ 2 is a fixed integer. The game is over as soon as a monochromatic copy of some fixed graph F has been created, and Painter’s goal is to ‘survive’ for as many steps as possible before this happens. We present a new technique for deriving upper bounds on the threshold of this game, i.e., on the typical number of steps Painter will survive with an optimal strategy. More specifically, we consider a deterministic two-player variant of the game where the edges are not chosen randomly, but by a second player Builder. However, Builder has to adhere to the restriction that, for some real number d, the ratio of edges to vertices in all subgraphs of the evolving board never exceeds d. We show that the existence of a winning strategy for Builder in this deterministic game implies an upper bound of n2−1/d for the threshold of the original probabilistic game. Moreover, we show that the best bound that can be derived in this way is indeed the threshold of the game if F is a forest. We illustrate these general results with several examples. The technique proposed here has been used by Balogh and Butterfield [2] to derive the first nontrivial upper bounds for the threshold of the game where F is a triangle and more than two colors are available.