Online Ramsey numbers of the claw versus cycles
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914242801696768 |
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| author | Zhi, Hexuan Zhang, Yanbo |
| author_facet | Zhi, Hexuan Zhang, Yanbo |
| contents | The online Ramsey number $\tilde r(G,H)$ is defined via a Builder--Painter game on an empty graph with countably many vertices. In each round, Builder reveals an edge, which Painter immediately colors either red or blue. Builder wins once a red copy of $G$ or a blue copy of $H$ appears, and $\tilde r(G,H)$ is the minimum number of edges Builder must reveal to force a win.
For a long cycle $C_\ell$, the online Ramsey numbers $\tilde r(G,C_\ell)$ are known only for a few specific choices of $G$. In particular, exact values were determined for $G=P_3$ by Cyman, Dzido, Lapinskas, and Lo (Electron. J. Combin., 2015), while asymptotically tight results were obtained when $G$ is an even cycle by Adamski, Bednarska-Bzdȩga, and Blažej (SIAM J. Discrete Math., 2024). In this paper, we consider the case where $G$ is the claw $K_{1,3}$ and determine the exact value of $\tilde r(K_{1,3},C_\ell)$. We show that \[ \tilde r(K_{1,3},C_\ell)=\left\lfloor \frac{3(\ell+1)}{2} \right\rfloor \quad \text{for all } \ell \ge 13. \] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_05452 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Online Ramsey numbers of the claw versus cycles Zhi, Hexuan Zhang, Yanbo Combinatorics 05C55, 05C57, 05D10 The online Ramsey number $\tilde r(G,H)$ is defined via a Builder--Painter game on an empty graph with countably many vertices. In each round, Builder reveals an edge, which Painter immediately colors either red or blue. Builder wins once a red copy of $G$ or a blue copy of $H$ appears, and $\tilde r(G,H)$ is the minimum number of edges Builder must reveal to force a win. For a long cycle $C_\ell$, the online Ramsey numbers $\tilde r(G,C_\ell)$ are known only for a few specific choices of $G$. In particular, exact values were determined for $G=P_3$ by Cyman, Dzido, Lapinskas, and Lo (Electron. J. Combin., 2015), while asymptotically tight results were obtained when $G$ is an even cycle by Adamski, Bednarska-Bzdȩga, and Blažej (SIAM J. Discrete Math., 2024). In this paper, we consider the case where $G$ is the claw $K_{1,3}$ and determine the exact value of $\tilde r(K_{1,3},C_\ell)$. We show that \[ \tilde r(K_{1,3},C_\ell)=\left\lfloor \frac{3(\ell+1)}{2} \right\rfloor \quad \text{for all } \ell \ge 13. \] |
| title | Online Ramsey numbers of the claw versus cycles |
| topic | Combinatorics 05C55, 05C57, 05D10 |
| url | https://arxiv.org/abs/2601.05452 |