Economical extremal hypergraphs for the Erdős-Selfridge theorem

A positional game can be thought of as a generalization of Tic-Tac-Toe played on a hypergraph (V,H). We study the Maker-Breaker game in which Maker wins if she occupies all of the vertices in an edge of H; otherwise Breaker wins. The Erdős–Selfridge Theorem, a significant result in positional game theory, gives criteria for the existence of an explicit winning strategy for Breaker for the game played on H. The bound in this theorem has been shown to be tight, as there are several examples of extremal hypergraphs for this theorem. We focus on the n-uniform extremal hypergraphs on which Maker has an economical (n-turn) winning strategy. We prove two distinct characterizations of these economical extremal hypergraphs.