2026/07/28 by Zoltán Lóránt Nagy
Mathematics · #math.CO
11 pages
arxiv created 2026/07/28 · arxiv updated 2026/07/30
Alon and Füredi proved that at least n affine hyperplanes are required to cover \0,1\n∖\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 maxH∈\mathcal H |H∩\0,1\n| over all families H of n affine hyperplanes covering \0,1\n∖0 and avoiding the origin. We give an explicit construction, proving that F(n)=(1+o(1))(2n)/(n), and hence asymptotically attain the averaging lower bound.