The Picker–Chooser diameter game

Positional Games are played under several rules on the same hypergraph. We consider some intriguing connections among the outcomes of the Maker–Breaker and the Picker–Chooser versions. The latter one was introduced by Beck (2002) [5], and proved to be important in understanding Positional Games in general. Beck had the profound conjecture that playing on the same hypergraph, Picker has better chances than Maker. The main goal of this paper is to confirm this conjecture for the notoriously hard diameter-2 game that was studied by Balogh et al. (2009) [1]. The diameter-2 game is also an example of the fact that the probabilistic intuition, or “Erdős paradigm” can fail completely, however, the acceleration of the game can restore it. The Picker–Chooser version is closer to Erdős paradigm: there are almost matching lower and upper bounds for the critical density. Pursuing these goals, we extend the theory of Picker–Chooser games to biased and discrepancy games, and develop Erdős–Selfridge type results.