A positional game is essentially a generalization of Tic-Tac-Toe played on a hypergraph?$(V,{cal F}).$ ?A pivotal result in the study of positional games is the?Erd?s–Selfridge?theorem, which?gives a simple criterion for the existence of a Breaker's winning strategy on a finite hypergraph ${cal F}$. ?It has been shown that the?bound in the?Erd?s–Selfridge?theorem can be tight and that numerous extremal hypergraphs?exist that demonstrate the tightness of the bound. We?focus on a generalization of the Erd?s–Selfridge?theorem proven by Beck for biased $(p:q)$?games,?which we call the $(p:q)$–Erd?s–Selfridge?theorem. ?We show that for $pn$-uniform hypergraphs there is a unique extremal hypergraph for the $(p:q)$–Erd?s–Selfridge theorem when $qgeq 2$.
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机译：位置游戏本质上是在超图上玩的井字游戏的泛化？$（V，{ cal F}）。$？位置游戏研究的关键结果是“ Erd？s–Selfridge”定理。 ，这为有限超图$ { cal F} $上存在Breaker的获胜策略提供了简单的标准。已经证明，Erd-Selfridge定理中的边界可能是紧密的，并且存在大量极值超图，证明了边界的紧密性。我们集中在贝克证明的偏向$（p：q）$游戏的Erd-s-Selfridge定理的推广上，我们称之为$（p：q）$-Erd？s-Selfridge定理。 。 ？我们证明，对于$ pn $一致的超图，当$ q geq 2 $时，$（p：q）$ – Erd？s–Selfridge定理有一个唯一的极值超图。
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