On-line Ramsey Numbers

Consider the following game between two players, Builder and Painter. Builder draws edges one at a time and Painter colours 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 r̃(G) of G. Our main result, relating to the conjecture that r̃(Kt) = o( ( r(t) 2 ) ), is that there exists a constant c > 1 such that r̃(Kt) ≤ c−t ( r(t) 2 ) for infinitely many values of t. We also prove a more specific upper bound for this number, showing that there exists a constant c such that r̃(Kt) ≤ t−c log t log log t 4. Finally, we prove a new upper bound for the on-line Ramsey number of the complete bipartite graph Kt,t.