Determine the asymptotic value of the Maker–Breaker degree game

Determine the correct asymptotic value of the Maker–Breaker degree game, including whether the minimum degree that Breaker can guarantee on every finite hypergraph of rank r and minimum degree d has an asymptotic value beyond the established lower bound d/(r+1)−o(d); in particular, resolve the substantial gap between the d/3−o(d) lower bound and the natural d/2 upper bound for d-regular graphs.

Background

The paper studies the Maker–Breaker degree game on finite hypergraphs: Maker and Breaker alternately claim hyperedges, and Breaker aims to maximize the minimum degree of the spanning subhypergraph formed by his claimed edges. The main theorem proves that for fixed rank r and sufficiently large minimum degree d, Breaker can guarantee minimum degree at least d/(r+1)−√(d log d). For graphs, this yields a d/3−√(d log d) guarantee, improving the classical d/4 bound.

The authors identify the correct asymptotic value of this game as the main unresolved issue. Even for d-regular graphs, the proven lower bound is asymptotically d/3, whereas a natural upper bound is d/2, leaving a substantial interval in which the true asymptotic value could lie.