Three-color online Ramsey numbers $\tilde{r}(P_3,P_3,P_{\ell})$ and $\tilde{r}(P_3, P_3, C_{\ell})$

Abstract

For given graphs $G_1, \ldots, G_k$, let $\tilde{r}(G_1, \ldots, G_k)$ denote their online Ramsey number. In an influential paper on the online Ramsey numbers for paths and cycles, Cyman, Dzido, Lapinskas, and Lo (Electron. J. Combin., 2015) determined the exact values of $\tilde{r}(P_3, P_{\ell})$ and $\tilde{r}(P_3, C_{\ell})$. They also conjectured the exact value of $\tilde{r}(P_4, P_{\ell})$ and the limit of $\tilde{r}(P_k, P_{\ell})/\ell$ as $\ell \to \infty$ for $k \ge 5$. The former conjecture was independently confirmed by Bednarska-Bzd\c{e}ga (European J. Combin., 2024) and Y.B. Zhang and Y.X. Zhang (arXiv:2302.13640), while the latter was disproved by Mond and Portier (European J. Combin., 2024). In this paper, we extend this line of research to the three-color setting and establish the exact value of $\tilde{r}(P_3,P_3,P_{\ell})$ for $\ell\ge 2$ and $\tilde{r}(P_3, P_3, C_{\ell})$ for $\ell \ge 16$.