Proof of a Conjecture on Online Ramsey Numbers of Paths

For two graphs $G_1$ and $G_2$, the online Ramsey number $\tilde{r}(G_1,G_2)$ is the smallest number of edges that Builder draws on an infinite empty graph to guarantee that there is either a red copy of $G_1$ or a blue copy of $G_2$, under the condition that Builder draws one edge in each round and Painter immediately colors it red or blue. For online Ramsey numbers of paths, Cyman, Dzido, Lapinskas, and Lo conjectured that $\tilde{r}(P_4,P_{\mathrm{\ell }+1})=\lceil (7\mathrm{\ell }+2)\mathrm{/}5\rceil$ for all $\mathrm{\ell }\ge 3$ [Electron. J. Combin. 22 (2015) #P1.15]. We verify the conjecture in this paper.