Proof of a Conjecture on Online Ramsey Numbers of Stars versus Paths

Given two graphs $G$ and $H$, the online Ramsey number $\tilde{r}(G,H)$ is defined to be the minimum number of rounds that Builder can always guarantee a win in the following $(G, H)$-online Ramsey game between Builder and Painter. Starting from an infinite set of isolated vertices, in each round Builder draws an edge between some two vertices, and Painter immediately colors it red or blue. Builder's goal is to force either a red copy of $G$ or a blue copy of $H$ in as few rounds as possible, while Painter's goal is to delay it for as many rounds as possible. Let $K_{1,3}$ denote a star with three edges and $P_{\ell}$ a path with $\ell$ vertices. Latip and Tan conjectured that $\tilde{r}(K_{1,3}, P_{\ell})=(3/2+o(1))\ell$ [Bull. Malays. Math. Sci. Soc. 44 (2021) 3511--3521]. We show that $\tilde{r}(K_{1,3}, P_{\ell})=\lfloor 3\ell/2 \rfloor$ for $\ell\ge 2$, which verifies the conjecture in a stronger form.