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

Abstract: 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 $δ(H_{\mathrm B})$ of his spanning subhypergraph $H_{\mathrm B}$. We prove that, for every fixed (r\ge2) and all sufficiently large (d), Breaker has a deterministic strategy satisfying [ δ(H_{\mathrm B})\ge \frac{d}{r+1}-\sqrt{d\log d}. ] For graphs, this improves the classical universal lower bound (d/4) to (d/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.