A Counterexample to Affine-Subspace Union Stability under Small Deletions
We construct a family of affine coordinate subspaces of $\mathbb{F}_2^N$ whose union is the entire space, but for which deleting a $1/N$ fraction from each member removes a fraction larger than $N^{-c}$ from the union, for every fixed $c>0$ and arbitrarily large $N$. Each subspace has codimension $\log_2 N$, and the family has at most $2N^2$ members. The construction uses all monochromatic coordinate conditions of size $k$ on $2k$ distinguished coordinates, where $N=2^k$, and deletes their common intersection with the middle Hamming layer. The exact global loss is $\binom{2k}{k}/4^k$. This…
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