On-line Ramsey Numbers of Paths and Cycles

Consider a game played on the edge set of the infinite clique by two players, Builder and Painter. In each round, Builder chooses an edge and Painter colours it red or blue. Builder wins by creating either a red copy of G or a blue copy of H for some fixed graphs G and H. The minimum number of rounds within which Builder can win, assuming both players play perfectly, is the on-line Ramsey number r̃(G,H). In this paper, we consider the case where G is a path Pk. We prove that r̃(P3, P`+1) = d5`/4e = r̃(P3, C`) for all ` > 5, and determine r̃(P4, P`+1) up to an additive constant for all ` > 3. We also prove some general lower bounds for on-line Ramsey numbers of the form r̃(Pk+1, H).