Improved Upper Bounds for Slicing the Hypercube

A collection of hyperplanes H slices all edges of the n-dimensional hypercube Qn with vertex set -1,1n 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 Qn. We prove that S(n) ≤ lceil frac4n5 rceil, except when n is an odd multiple of 5, in which case S(n) ≤ frac4n5 +1. This improves upon the previously known upper bound of S(n) ≤ lceilfrac5n6 rceil due to Paterson reported in 1971. We also obtain new lower bounds on the maximum number of edges in Qn that can be sliced using k<n hyperplanes. We prove the improved upper bound on S(n) by constructing 8 hyperplanes slicing Q10 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.