AI 中文总结
该研究针对Alon-Füredi覆盖问题，构造出显式的n个仿射超平面族，证明其最大布尔交集的最小值渐近达到平均下界，解决了覆盖的超平面规模与布尔交集大小的相关问题。
AI 中文摘要
Alon和Füredi证明，要覆盖{0,1}^n\{0}且避开原点，至少需要n个仿射超平面，且该下界是紧的。我们研究达到这个最小值的覆盖中，超平面间最大布尔交集的最小可能值。设F(n)为所有覆盖{0,1}^n\{0}且避开原点的n个仿射超平面族ℋ中，max_{H∈ℋ}|H∩{0,1}^n|的最小值。我们给出显式构造，证明F(n)=(1+o(1))2^n/n，从而渐近达到平均下界。
英文摘要
Alon and Füredi proved that at least $n$ affine hyperplanes are required to cover $\{0,1\}^n\setminus\{\textbf{0}\}$ while avoiding the origin, and that this bound is sharp. We study how small the largest Boolean intersection among the hyperplanes can be in a cover attaining this minimum. Let $F(n)$ denote the minimum possible value of $\max_{H\in\mathcal H} |H\cap\{0,1\}^n|$ over all families $\mathcal{H}$ of $n$ affine hyperplanes covering $\{0,1\}^n\setminus{\mathbf{0}}$ and avoiding the origin. We give an explicit construction, proving that $F(n)=(1+o(1))\frac{2^n}{n},$ and hence asymptotically attain the averaging lower bound.
Comments11 pages
