Upper bounds on positional Paris–Harrington games

We give upper bounds for a positional game – in the sense of Beck – based on the Paris–Harrington Theorem for bi-colorings of graphs and uniform hypergraphs of arbitrary dimension. The bounds for the positional game show a striking difference when compared to the bounds for 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 3-uniform hypergraphs are a fixed stack of exponentials while the bounds for the corresponding combinatorial principle are known to be at least Ackermannian. For higher dimensions, the combinatorial Paris–Harrington numbers are known to be cofinal in the Schwichtenberg–Wainer Hierarchy of fast-growing functions up to the ordinal $arepsilon_0$, while we show that the game Paris–Harrington numbers are bounded by fixed stacks of exponentials.