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[2602.16807] Improved Upper Bounds for Slicing the Hypercube
Summary
This paper presents improved upper bounds for slicing the edges of the n-dimensional hypercube using hyperplanes, enhancing previous research and introducing new mathematical tools.
Why It Matters
Understanding the slicing of hypercubes has implications in various fields such as combinatorics and artificial intelligence. The improved bounds can lead to advancements in algorithm design and mathematical reasoning, particularly with the integration of automated tools like CPro1.
Key Takeaways
- The minimum number of hyperplanes needed to slice the n-dimensional hypercube is improved to S(n) ≤ ⌈4n/5⌉.
- The new bounds are significant for odd multiples of 5, where S(n) ≤ (4n/5) + 1.
- The research utilizes CPro1, an automated tool that leverages reasoning LLMs for mathematical discovery.
- This work builds on previous findings from 1971, showcasing advancements in the field.
- New lower bounds for the maximum number of edges sliced by k hyperplanes are also established.
Computer Science > Artificial Intelligence arXiv:2602.16807 (cs) [Submitted on 6 Feb 2026] Title:Improved Upper Bounds for Slicing the Hypercube Authors:Duncan Soiffer, Nathaniel Itty, Christopher D. Rosin, Blake Bruell, Mason DiCicco, Gábor N. Sárközy, Ryan Offstein, Daniel Reichman View a PDF of the paper titled Improved Upper Bounds for Slicing the Hypercube, by Duncan Soiffer and 7 other authors View PDF HTML (experimental) 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. Subjects: Artificial Intelligence (cs.AI); Discrete Mathema...
