We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph on n vertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when Proposer has no legal moves remaining, and Proposer wins if the final graph has independence number at least s. We prove a threshold phenomenon exists for this game by exhibiting randomized strategies for both players that are optimal up to constants. Namely, there exist constants 0 < A < B such that (under optimal play) Proposer wins with high probability if , while Decider wins with high probability if . This is a factor of larger than the lower bound coming from the off‐diagonal Ramsey number r(3,s).

中文翻译：

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拉姆西，纸，剪刀

我们介绍一种称为Ramsey，Paper和Scissors的图Ramsey游戏。该游戏有两名玩家，提议者和决定者。从n个顶点上的空图开始，每转一圈Proposer提出一个潜在边，Decider同时决定（不知道Proposer的选择）是否将其添加到图中。提议者无法提出将在图形中创建三角形的边。当提议者没有合法移动余地时，游戏结束；如果最终图形的独立性数至少为s，则提议者获胜。我们通过对两个玩家展示出最高达常数的随机策略，证明该游戏存在阈值现象。即，存在常数0 <  A  <  B这样，（在最佳玩法下），如果，提议者以高概率获胜，而决策者则以高概率获胜。这是一个比非对角Ramsey数r（3，s）的下界更大的因数。