AI 中文总结
该研究针对Maker--Breaker度博弈问题，通过引入虚拟平衡过程、乘性风险编码及维持风险向量在Shearer区域的策略，证明了秩r超图下Breaker可实现最小度下界d/(r+1)−√(d log d)，将图情形的经典下界从d/4提升至d/3−√(d log d)。
AI 中文摘要
设H为秩r、最小度d的有限超图。在Maker--Breaker度博弈中，Maker和Breaker交替认领H中未被认领的超边，Maker先手，Breaker的目标是最大化其生成子超图H_B的最小度δ(H_B)。我们证明，对任意固定的r≥2以及所有足够大的d，Breaker存在确定性策略满足δ(H_B)≥d/(r+1)−√(d log d)。对于图的情形，该结果将经典的通用下界d/4提升至d/3−√(d log d)。证明引入了一种虚拟平衡过程，通过乘性风险编码局部不平衡，并在博弈全程将所得风险向量维持在Shearer区域内。
英文摘要
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.
