Overview
This research presents updated upper bounds for $S(n)$, which denotes the minimum number of hyperplanes required to slice all edges of an $n$-dimensional hypercube, $Q_n$. The hypercube's vertices are defined as $\{-1,1\}^n$. A collection of hyperplanes $\mathcal{H}$ is considered to slice all edges of $Q_n$ if every edge $e$ within the hypercube is intersected in its interior by at least one hyperplane in $\mathcal{H}$.
Research Context
The problem of slicing the hypercube $Q_n$ has been previously investigated, with a notable upper bound established by Paterson in 1971, stating $S(n) \leq \lceil\frac{5n}{6} \rceil$. This current work builds upon this long-standing result by seeking to improve the upper bound for $S(n)$.
Approach
The researchers developed improved upper bounds for $S(n)$ by constructing hyperplanes. A key aspect of their methodology involved the use of an automatic tool named CPro1. This tool integrates reasoning Large Language Models (LLMs) with automated hyperparameter tuning to generate search algorithms. These algorithms were then utilized for the discovery of mathematical constructions. Specifically, CPro1 aided in the construction of 8 hyperplanes that slice $Q_{10}$, which informed the derivation of the general upper bound.
Findings
- The research established a new upper bound for $S(n)$: $S(n) \leq \lceil \frac{4n}{5} \rceil$.
- An exception to this general bound applies when $n$ is an odd multiple of $5$, in which case the upper bound is $S(n) \leq \frac{4n}{5} +1$.
- These new upper bounds represent an improvement over the previously known bound of $S(n) \leq \lceil\frac{5n}{6} \rceil$, which was reported by Paterson in 1971.
- Additionally, new lower bounds were obtained regarding the maximum number of edges in $Q_n$ that can be sliced utilizing $k n$ hyperplanes.
- The specific construction of 8 hyperplanes slicing $Q_{10}$ was achieved with the assistance of the CPro1 tool.