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 r~(G,H). In this paper, we consider the case
where G is a path Pkβ. We prove that r~(P3β,Pβ+1β)=β5β/4β=r~(P3β,Cββ) for all ββ₯5, and determine
r~(P4β,Pβ+1β) up to an additive constant for all ββ₯3.
We also prove some general lower bounds for on-line Ramsey numbers of the form
r~(Pk+1β,H).Comment: Preprin