Maker-Breaker is solved in polynomial time on hypergraphs of rank 3
This paper investigates the winner determination problem for Maker-Breaker positional games on 3-uniform hypergraphs. While the problem is PSPACE-complete on general 5-uniform hypergraphs, polynomial-time algorithms were previously known only for two restricted subclasses; Rahman and Watson (2020) conjectured tractability for all 3-uniform hypergraphs. We confirm this conjecture by introducing a “vertex hazard” analytical framework and defining the novel notion of “hazardous subhypergraphs.” We construct a critical family ℱ of hazardous sets and establish a structural characterization: Breaker wins if and only if, at every vertex, all ℱ-hazardous sets pairwise intersect. Based on this, we design the first polynomial-time algorithm for arbitrary 3-uniform hypergraphs, reducing the complexity from PSPACE to P. Furthermore, we prove that if Maker wins, she can achieve her goal within O(log n) moves, and we correct an erroneous claim in recent literature.