Improved Upper Bounds for Slicing the Hypercube

2026-02-06

arXiv.org

AI Summary

This study addresses the problem of determining the minimum number of hyperplanes, denoted $S(n)$, required to slice all edges of an $n$-dimensional hypercube. By integrating reasoning large language models with CPro1—a tool featuring automated hyperparameter tuning—we design an efficient search algorithm that yields novel constructive solutions, including an 8-hyperplane slicing scheme for the 10-dimensional hypercube ($Q_{10}$). Our main contributions are a significant improvement of the upper bound on $S(n)$ from $\lceil 5n/6 \rceil$ to $\lceil 4n/5 \rceil$, with a refined bound of $4n/5 + 1$ when $n$ is an odd multiple of 5, and the first non-trivial lower bound on the number of edges that can be sliced by fewer than $n$ hyperplanes. These results substantially advance the theoretical understanding of this classical problem in combinatorial geometry.

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 $S(n) \leq \lceil\frac{5n}{6} \rceil$ 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.

Problem

• hypercube
• slicing
• hyperplanes
• edge covering
• combinatorial geometry

Innovation

• hypercube slicing
• upper bound improvement
• automated mathematical discovery
• reasoning LLMs
• CPro1