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 $\delta(H_{\mathrm B})$ 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 \[ \delta(H_{\mathrm B})\ge \frac{d}{r+1}-\sqrt{d\log d}. \] 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.