Improved Upper Bounds for Slicing the Hypercube

arXiv:2602.16807v1 Announce Type: new
Abstract: A collection of hyperplanes $mathcal{H}$ slices all edges of the $n$-dimensional hypercube $Q_n$ with vertex set ${-1,1}^n$ if, for every edge $e$ in the hypercube, there exists a hyperplane in $mathcal{H}$ intersecting $e$ in its interior. Let $S(n)$ be the minimum number of hyperplanes needed to slice $Q_n$. We prove that $S(n) leq lceil frac{4n}{5} rceil$, except when $n$ is an odd multiple of $5$, in which case $S(n) leq frac{4n}{5} +1$. This improves upon the previously known upper bound of $S(n) leq lceilfrac{5n}{6} rceil$ due to Paterson reported in 1971. We also obtain new lower bounds on the maximum number of edges in $Q_n$ that can be sliced using $k