Improved Upper Bounds for Slicing the Hypercube

arXiv:2602.16807v2 Announce Type: replace Abstract: A collection of hyperplanes 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 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<n hyperplanes. We prove the improved upper bound on S(n) by constructing 8 hyperplanes slicing Q_{10} aided by the recently introduced CPro1: an automatic tool that uses reasoning LLMs coupled with automated hyperparameter tuning to create search algorithms for the discovery of mathematical constructions.

Source: arXiv cs.AI | 2026-07-01