Improved Upper Bounds for Slicing the Hypercube

TL;DR

This paper improves the upper bounds on the minimum number of hyperplanes needed to slice all edges of an n-dimensional hypercube, using a novel construction aided by automated reasoning tools, and provides new lower bounds on sliced edges.

Contribution

It presents improved upper bounds for hypercube slicing and introduces an automated reasoning approach to construct slicing hyperplanes.

Findings

Upper bound for S(n) is now rac{4n}{5}, with a slight exception for multiples of 5.

Constructed slicing hyperplanes for Q_{10} using automated reasoning tools.

Established new lower bounds on the maximum edges sliced by fewer than n hyperplanes.

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.