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

TL;DR

This paper determines the exact three-color online Ramsey numbers for paths and cycles, extending previous two-color results and addressing conjectures in the field of combinatorics.

Contribution

It establishes the exact values of  r~(P_3,P_3,P_{}) and  r~(P_3, P_3, C_{}) for specific parameters, advancing understanding of multi-color online Ramsey numbers.

Findings

Exact value of  r~(P_3,P_3,P_{}) for   2.

Exact value of  r~(P_3, P_3, C_{}) for   16.

Extension of two-color results to three-color setting.

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 $\ell \ge 16$.