On-line Ramsey Numbers

Consider the following game between two players, Builder and Painter. Builder draws edges one at a time and Painter colors them in either red or blue, as each appears. Builder's aim is to force Painter to draw a monochromatic copy of a fixed graph $G$. The minimum number of edges which Builder must draw, regardless of Painter's strategy, in order to guarantee that this happens is known as the on-line Ramsey number $\tilde{r}(G)$ of $G$. Our main result, relating to the conjecture that $\tilde{r}(K_t)=o(({r(t)\atop2}))$, is that there exists a constant $c>1$ such that $\tilde{r}(K_t)\leq c^{-t}({r(t)\atop2})$ for infinitely many values of $t$. We also prove a more specific upper bound for this number, showing that there exists a positive constant $c$ such that $\tilde{r}(K_t)\leq t^{-c\frac{\log t}{\log \log t}}4^t$. Finally, we prove a new upper bound for the on-line Ramsey number of the complete bipartite graph $K_{t,t}$.