Three-color online Ramsey numbers $\tilde{r}(P_3,P_3,P_{\ell})$ and $\tilde{r}(P_3, P_3, C_{\ell})$
Hexuan Zhi, Yanbo Zhang
TL;DR
This work resolves exact values for three-color online Ramsey numbers in the sparse-path/cycle regime: the problems \tilde{r}(P_3,P_3,P_ℓ) and \tilde{r}(P_3,P_3,C_ℓ). Building on the Builder–Painter framework used for two colors, it combines adversarial lower-bound constructions with explicit, modular-structured upper-bound strategies. The results give precise thresholds: for ℓ≥8 with ℓ≡0 (mod 4), \tilde{r}(P_3,P_3,P_ℓ)=3ℓ/2−1 and otherwise ⌊3ℓ/2⌋; for ℓ≥16 with ℓ≡3 (mod 4), \tilde{r}(P_3,P_3,C_ℓ)=(3ℓ+1)/2 and otherwise ⌊3(ℓ+1)/2⌋. The proofs employ unit-graph constructions, path- and cycle-extension lemmas, and modular-case analyses to achieve tight, sometimes exact, bounds, contributing a refined understanding of online Ramsey dynamics for sparse graphs under three colors.
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-Bzdȩga (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 $\ell \ge 16$.