Computational complexity of rank-4 Maker-Breaker games

Determine the computational complexity of deciding the winner of Maker-Breaker games played on hypergraphs of rank 4.

Background

The paper establishes that deciding the winner of Maker-Breaker games is PSPACE-complete for 5-uniform hypergraphs. It also notes that Maker-Breaker games on hypergraphs of rank 3 or lower are solvable in polynomial time. The complexity of the intervening rank-4 case is therefore left unresolved, forming the natural boundary problem between the known polynomial-time and PSPACE-complete results.

References

This leaves us with an obvious open problem: How difficult is solving Maker-Breaker games on hypergraphs of rank 4?

— Solving Maker-Breaker Games on 5-uniform hypergraphs is PSPACE-complete  (2502.20271 - Koepke, 27 Feb 2025) in Section 7, Conclusion, p. 12

The complexity of the usual edge-based C_4-game remains open.

— A Linear-Time Algorithm for the Unbiased Claw Game on Graphs with Maximum Degree 4  (2610.21878 - Brinkmann, 15 Oct 2026) in Section 1, Introduction