In an influential paper, Erd\H{o}s and Selfridge introduced the Maker-Breaker game played on a hypergraph, or equivalently, on a monotone CNF. The players take turns assigning values to variables of their choosing, and Breaker's goal is to satisfy the CNF, while Maker's goal is to falsify it. The Erd\H{o}s-Selfridge Theorem says that the least number of clauses in any monotone CNF with $k$ literals per clause where Maker has a winning strategy is $\Theta(2^k)$. We study the analogous question when the CNF is not necessarily monotone. We prove bounds of $\Theta(\sqrt{2}\,^k)$ when Maker plays last, and $\Omega(1.5^k)$ and $O(r^k)$ when Breaker plays last, where $r=(1+\sqrt{5})/2\approx 1.618$ is the golden ratio.
翻译:在有影响力的报纸《Erd\H{o}s & Selfridge 》 中, 引入了在高光或等效的单调 CNF 上玩的Maker-Breaker游戏。 玩家轮流将值分配给他们选择的变量, 断开器的目标是满足 CNF, 而 Maker 的目标是伪造它。 Erd\H{o}s- Ofiterridge Theorem 表示, 任何单调 CNF 中, 以每条款中以$k$( 立方美元) 来赢得策略的单调中, 最小的条款数量为$\ Theta( 2 ⁇ k) 。 当 CNF 不一定是单调时, 我们研究类似的问题。 我们证明了当 Maker 玩家最后一场游戏时$\theta( sqr{% 2 ⁇, k) 和$O( r) $( $) (r= ( {sqr{5})/2\ approx 1. 1.618$) 是黄金比率 。