A Note on On-Line Ramsey Numbers of Stars and Paths

Abstract

An on-line Ramsey game is a game between two players, Builder and Painter, on an initially empty graph with unbounded set of vertices. In each round, Builder selects two vertices and joins them with an edge while Painter colours the edge immediately with either red or blue. Builder’s aim is to force either a red copy of a fixed graph G or a blue copy of a fixed graph H. The game ends with Builder’s victory when Builder manages to force either a red G or a blue H. The minimum number of rounds for Builder to win the game, regardless of Painter’s strategy, is the on-line Ramsey number $${\tilde{r}}(G,H)$$ . This note focuses on the case when G and H are stars and paths. In particular, we will prove the upper bound of $${\tilde{r}}(S_3,P_{l+1})\le 5l/3+2$$ . We will also present an alternative proof for the upper bound of $${\tilde{r}}(S_2 = P_3, P_{l+1}) = \lceil 5l/4\rceil $$ .