Unified threatening-structure framework for rank-three Maker–Breaker games

Develop a unified view of the threatening-structure technique for 3-uniform hypergraphs, with the aim of obtaining improved algorithms for Maker–Breaker games on hypergraphs of rank at most three.

Background

The paper's analysis exploits vertices of degree four as forcing or threatening structures in the unbiased claw game, in a manner analogous to the use of copies of K4 in the unbiased triangle game. The authors ask whether these game-specific ideas can be subsumed under a common framework for 3-uniform hypergraphs.

A successful unification could improve algorithms for Maker–Breaker games on hypergraphs of rank at most three, beyond the techniques used for the claw and triangle games separately.