Upper Bounds for Positional Paris-Harrington Games

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.