AI 中文总结
本文研究可数无限完全图上带预着色边的Maker-Breaker游戏，针对保色与保模式两类游戏，分别给出获胜条件、完整刻画及Maker有胜策的模式，部分解答了相关学者的问题。
AI 中文摘要
给定图$B$和$G$，在经典的Maker-Breaker游戏$\text{MB}(B,G)$中，两名玩家Maker和Breaker轮流占领$B$的边，Maker的目标是在$B$中占领一个$G$的副本，而Breaker的目标是阻止Maker达成该目标。在本文中，$B$是可数无限完全图$K_{\boldsymbol{/aleph_0}}$，且给定有限多个无限子图$G_1, \boldsymbol{/dots}, G_k \boldsymbol{/subseteq} B$。在保色游戏中，Maker的目标是占领一个$K_{\boldsymbol{/aleph_0}} \boldsymbol{/subseteq} B$，该图包含每个$G_i$的无限多条边。对于$k>1$的情况，我们给出了Maker和Breaker各自的充分获胜条件；对于$k=1$的情况，我们对该游戏进行了完整刻画。这部分回答了Bowler、Emde和Gut提出的一个问题。在（部分）保模式游戏中，Maker的目标是占领一个$K_{\boldsymbol{/aleph_0}}$的副本$K$，使得对所有$i \boldsymbol{/in} [k]$，$G_i \boldsymbol{/cap} K$同构于$G_i$（或$G_i$的一个子图）。在这类游戏中，我们研究了一些Maker具有获胜策略的模式。
英文摘要
Suppose we are given graphs $B$ and $G$. In the classical Maker-Breaker game $\text{MB}(B,G)$ two players, Maker and Breaker, alternately claim edges of $B$ and it is Maker's goal to claim a copy of $G$ in $B$, while it is Breaker's goal to prevent that. In this paper, $B$ is the countably infinite complete graph $K_{\aleph_0}$ and we are given finitely many infinite subgraphs $G_1, \dots, G_k \subseteq B$. In the color preserving game, it will be Maker's goal to claim a $K_{\aleph_0} \subseteq B$, which contains infinitely many edges of each $G_i$. We present sufficient winning conditions for both Maker and Breaker, if $k > 1$ and a full characterization of the game, if $k =1$. This partly answers a question of Bowler, Emde and Gut. In the (partially) pattern preserving game, it is Maker's goal to claim a copy $K$ of $K_{\aleph_0}$, such that $G_i \cap K$ is isomorphic to (a subgraph of) $G_i$ for all $i \in [k]$. In those games, we investigate some patterns for which Maker has a winning strategy.
Comments26 pages, 10 figures
