Online Ramsey Numbers and the Subgraph Query Problem

The (m,n)-online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge on each turn and Painter immediately paints it red or blue. Builder's goal is to force Painter to create either a red K_m or a blue K_n using as few turns as possible. The online Ramsey number [equation; see abstract in PDF for details] is the minimum number of edges Builder needs to guarantee a win in the (m,n)-online Ramsey game. By analyzing the special case where Painter plays randomly, we obtain an exponential improvement [equation; see abstract in PDF for details] for the lower bound on the diagonal online Ramsey number, as well as a corresponding improvement [equation; see abstract in PDF for details] for the off-diagonal case, where m ≥ 3 is fixed and n → ∞. Using a different randomized Painter strategy, we prove that [equation; see abstract in PDF for details], determining this function up to a polylogarithmic factor. We also improve the upper bound in the off-diagonal case for m ≥ 4. In connection with the online Ramsey game with a random Painter, we study the problem of finding a copy of a target graph H in a sufficiently large unknown Erdős-Rényi random graph G(N,p) using as few queries as possible, where each query reveals whether or not a particular pair of vertices are adjacent. We call this problem the Subgraph Query Problem. We determine the order of the number of queries needed for complete graphs up to five vertices and prove general bounds for this problem.