Extremal Hypergraphs for the Biased Erdős-Selfridge Theorem

A positional game is essentially a generalization of Tic-Tac-Toe played on a hypergraph (V,F). A pivotal result in the study of positional games is the Erdős– Selfridge theorem, which gives a simple criterion for the existence of a Breaker’s winning strategy on a finite hypergraph F . It has been shown that the bound in the Erdős–Selfridge theorem can be tight and that numerous extremal hypergraphs exist that demonstrate the tightness of the bound. We focus on a generalization of the Erdős–Selfridge theorem proven by Beck for biased (p : q) games, which we call the (p : q)–Erdős–Selfridge theorem. We show that for pn-uniform hypergraphs there is a unique extremal hypergraph for the (p : q)–Erdős–Selfridge theorem when q > 2.