A One-Third Bound for the Maker--Breaker Degree Game

Let $H$ be a finite hypergraph with rank $r$ and minimum degree $d$. In the Maker–Breaker degree game, Maker and Breaker alternately claim previously unclaimed hyperedges of $H$, with Maker moving first, and Breaker seeks to maximize the minimum degree $\delta (H_{\mathrm{B}})$ of his spanning subhypergraph $H_{\mathrm{B}}$. We prove that, for every fixed $r\ge 2$ and all sufficiently large $d$, Breaker has a deterministic strategy satisfying $$\delta (H_{\mathrm{B}})\ge \frac{d}{r+1}-\sqrt{d\log d}\mathrm{.}$$ For graphs, this improves the classical universal lower bound $d\mathrm{/}4$ to $d\mathrm{/}3-\sqrt{d\log d}$. The proof introduces a virtual balancing process, encodes local imbalance by a multiplicative risk, and keeps the resulting risk vector in the Shearer region throughout the game.