We give upper bounds for a positional game — in the sense of Beck — based on the Paris-Harrington principle for bi-colorings of graphs and uniform hypergraphs of arbitrary dimension. The bounds show a striking difference with respect to the bounds of the combinatorial principle itself. Our results confirm a phenomenon already observed by Beck and others: the upper bounds for the game version of a combinatorial principle are drastically smaller than the upper bounds for the principle itself. In the case of Paris-Harrington games the difference is qualitatively very striking. For example, the bounds for the game on 3uniform hypergraphs are a fixed stack of exponentials while the bounds on the corresponding combinatorial principle are known to be Ackermannian! For higher dimensions, the combinatorial Paris-Harrington numbers are known to be cofinal in the Schwichtenberg-Wainer Hiearchy of fast-growing functions up to ε0, while we show that the game Paris-Harrington numbers are fixed stacks of exponentials.

中文翻译：

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位置巴黎-哈灵顿游戏的上限

我们给出了位置博弈的上限——在贝克的意义上——基于巴黎-哈灵顿原则，用于图的双色和任意维度的统一超图。边界与组合原理本身的边界显示出显着的差异。我们的结果证实了 Beck 和其他人已经观察到的一个现象：组合原理的博弈版本的上限远小于该原理本身的上限。在巴黎-哈灵顿比赛的情况下，差异在性质上非常显着。例如，3uniform hypergraphs 上的游戏边界是一个固定的指数堆栈，而相应组合原理的边界已知是阿克曼！对于更高的维度，