Economical tight examples for the biased Erdős–Selfridge theorem Original Research Article

A positional game is essentially a generalization of tic-tac-toe played on a hypergraph image. A pivotal result in the study of positional games is the Erdős–Selfridge theorem, which gives simple criteria for the existence of a Breakerʹs winning strategy on a hypergraph image. It has been shown that the Erdős–Selfridge theorem can be tight and that numerous extremal systems exist for that theorem. We focus on a generalization of the Erdős–Selfridge theorem proven by Beck for biased image games, which we call the image–Erdős–Selfridge theorem. We show that for pn-uniform hypergraphs there is a unique extremal system for the image–Erdős–Selfridge theorem (image) when Maker must win in exactly n turns (i.e., as quickly as possible).