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 nonedges and is presented to the player, henceforth called Painter. Painter must assign one of $r$ available colors to each edge immediately, where $r\geq 2$ 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 chosen not 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 $n^{2-1/d}$ 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 [Discrete Math., 310 (2010), pp. 3653--3657] 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. [PUBLICATION ABSTRACT]