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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.06830 |
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| _version_ | 1866912205069352960 |
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| author | Ai, Jiangdong Gao, Jun Xu, Zixiang Yan, Xin |
| author_facet | Ai, Jiangdong Gao, Jun Xu, Zixiang Yan, Xin |
| contents | The strong Ramsey game $R(\mathcal{B}, H)$ is a two-player game played on a graph $\mathcal{B}$, referred to as the board, with a target graph $H$. In this game, two players, $P_1$ and $P_2$, alternately claim unclaimed edges of $\mathcal{B}$, starting with $P_1$. The goal is to claim a subgraph isomorphic to $H$, with the first player achieving this declared the winner. A fundamental open question, persisting for over three decades, asks whether there exists a graph $H$ such that in the game $R(K_n, H)$, $P_1$ does not have a winning strategy in a bounded number of moves as $n \to \infty$.
In this paper, we shift the focus to the variant $R(K_n \sqcup K_n, H)$, introduced by David, Hartarsky, and Tiba, where the board $K_n \sqcup K_n$ consists of two disjoint copies of $K_n$. We prove that there exist infinitely many graphs $H$ such that $P_1$ cannot win in $R(K_n \sqcup K_n, H)$ within a bounded number of moves through a concise proof. This perhaps provides evidence for the existence of examples to the above longstanding open problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06830 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strong Ramsey game on two boards Ai, Jiangdong Gao, Jun Xu, Zixiang Yan, Xin Combinatorics 05C57 The strong Ramsey game $R(\mathcal{B}, H)$ is a two-player game played on a graph $\mathcal{B}$, referred to as the board, with a target graph $H$. In this game, two players, $P_1$ and $P_2$, alternately claim unclaimed edges of $\mathcal{B}$, starting with $P_1$. The goal is to claim a subgraph isomorphic to $H$, with the first player achieving this declared the winner. A fundamental open question, persisting for over three decades, asks whether there exists a graph $H$ such that in the game $R(K_n, H)$, $P_1$ does not have a winning strategy in a bounded number of moves as $n \to \infty$. In this paper, we shift the focus to the variant $R(K_n \sqcup K_n, H)$, introduced by David, Hartarsky, and Tiba, where the board $K_n \sqcup K_n$ consists of two disjoint copies of $K_n$. We prove that there exist infinitely many graphs $H$ such that $P_1$ cannot win in $R(K_n \sqcup K_n, H)$ within a bounded number of moves through a concise proof. This perhaps provides evidence for the existence of examples to the above longstanding open problem. |
| title | Strong Ramsey game on two boards |
| topic | Combinatorics 05C57 |
| url | https://arxiv.org/abs/2501.06830 |