The online Ramsey game for graphs $G$ and $H$ is played on the infinite complete graph $K_\mathbb{N}$. Each round, Builder chooses an edge, and Painter colors it red or blue. The online Ramsey number $\tilde{r}(G,H)$ is the smallest integer $t$ for which Builder has a strategy that guarantees a red copy of $G$ or a blue copy of $H$ in at most $t$ rounds. We show that for every fixed $k$, there are constants $λ_1$ and $λ_2$ such that $\tilde{r}(P_k,P_n)/n$ and $\tilde{r}(P_k,C_n)/n$ converge to $λ_1$, and $\tilde{r}(K_{1,k},P_n)/n$ and $\tilde{r}(K_{1,k},C_n)/n$ converge to $λ_2$.

翻译：在线拉姆齐博弈涉及图$G$和$H$，在无限完全图$K_\mathbb{N}$上进行。每轮中，Builder选择一条边，Painter将其染成红色或蓝色。在线拉姆齐数$\tilde{r}(G,H)$是使得Builder存在策略、确保在至多$t$轮内出现红色$G$或蓝色$H$的最小整数$t$。我们证明：对每个固定整数$k$，存在常数$λ_1$和$λ_2$，使得$\tilde{r}(P_k,P_n)/n$和$\tilde{r}(P_k,C_n)/n$收敛于$λ_1$，而$\tilde{r}(K_{1,k},P_n)/n$和$\tilde{r}(K_{1,k},C_n)/n$收敛于$λ_2$。